"Woman, why are you crying? Who is it you are looking for?" (John 20:15)

Now Mary stood outside the tomb crying. As she wept, she bent over to look into the tomb and saw two angels in white, seated where Jesus' body had been, one at the head and the other at the foot. They asked her, "Woman, why are you crying?" "They have taken my Lord away," she said, "and I don't know where they have put him." At this, she turned around and saw Jesus standing there, but she did not realize that it was Jesus. He asked her, "Woman, why are you crying? Who is it you are looking for?" Thinking he was the gardener, she said, "Sir, if you have carried him away, tell me where you have put him, and I will get him." (John 20:11-20:15)

Why didn't Mary recognize Jesus?

As Mary is standing outside of Jesus' tomb - empty except for strips of linen - two angels appeared to her and then she turns around and Jesus appears.

But she doesn't recognize that it is Jesus:
At this, she turned around and saw Jesus standing there, but she did not realize that it was Jesus. (John 20:14)
Why didn't Mary - a close disciple of Jesus - "not realize that it was Jesus"

She was looking right at him. And she certainly knew what his body looked like. Why didn't she recognize him immediately?

The fact that Jesus wasn't recognized when Jesus appeared after his physical body died is described in the other Gospels too:
Afterward Jesus appeared in a different form to two of them while they were walking in the country. (Mark 16:12)
When they saw him, they worshiped him; but some doubted. (Matt. 28:17)
In the Book of Luke, Jesus appeared before Peter and another disciple, Cleopas, as they were walking:
Now that same day two of them were going to a village called Emmaus, about seven miles from Jerusalem. They were talking with each other about everything that had happened. As they talked and discussed these things with each other, Jesus himself came up and walked along with them; but they were kept from recognizing him. He asked them, "What are you discussing together as you walk along?" They stood still, their faces downcast. One of them, named Cleopas, asked him, "Are you the only one visiting Jerusalem who does not know the things that have happened there in these days?" "What things?" he asked. "About Jesus of Nazareth," they replied. "He was a prophet, powerful in word and deed before God and all the people. (Luke 24:13-19)
This last description of Jesus' appearance describes them speaking with Jesus further, while it is obvious they do not realize that it was Jesus.

Why didn't anyone recognize Jesus?

Why do all of these descriptions of Jesus' appearing to his disciples maintain that Jesus' close followers do not recognize him?

Because Jesus was not appearing to them in his physical body. 

What they knew as Jesus was his physical body. Jesus was appearing to them with what many would refer to as an apparition. That is, a manipulation of matter in such a way that makes a spirit-person appear in the physical world.

What appeared to them was Jesus' spiritual self, manifested within the physical world much as an angel will sometimes appear before the eyes of some people.

This is confirmed by the statement quoted above: "Jesus appeared in a different form."

What is meant by "different form"? It means a form different than the form they were used to seeing Jesus in - the physical body that had expired. As also mentioned with John 19:26-27, the reality of Jesus' different form is also confirmed in the ancient Nag Hammadi scripture called "The Wisdom of Jesus:"
After he rose from the dead body, his twelve disciples and seven women continued to follow him.... The Savior appeared not in his earlier form, but in the ethereal spirit. His appearance was like a beautiful illuminated angel. But I cannot properly describe his appearance. No material body could bear this – only the purified spiritual body – just as he taught us about from Galilee to the mount referred to as ‘the olives.’ (The Wisdom of Jesus 1-2)
Yes, Jesus' physical body died. But his spiritual self re-appeared to their physical eyes in a different form, just as had been predicted in the scriptures.

Didn't Jesus' body 'rise from the dead'?

Jesus did not "rise from the dead" as interpreted by many. Because Jesus never died. Only his physical body died.

This idea that Jesus' physical body "rose" is simply a myth. It has no basis in reality. Yes, Jesus' spiritual self "rose" from the body. But he then appeared to them "in a different form." His physical body did not "rise."

What about the linen they wrapped his body in, which was lying in the tomb? Doesn't that prove anything? Wouldn't that mean, if Jesus' body rose from the dead that he would be naked?

If anything, the remaining linen that clothed Jesus' body supports the notion that Jesus' physical body was taken from the tomb by the temple or Roman officials who themselves had feared this "rising from the dead" prophesy - or possibly even Joseph and his family. Jesus' body was, in fact, placed in a temporary tomb, as we'll illustrate below.

And yes one Gospel does describe how Jesus' disciples saw the hole in Jesus' side - specifically 'doubting Thomas.' After Thomas' challenge, Jesus let him put his hand through a hole in his side. What kind of physical body is this? A hole in his side? Where were the blood and the guts that physical bodies contain? The reality is that Jesus was able to manipulate what form his disciples could see him in. This is the power that angels also exhibit - given to them by the Supreme Being - to appear within the physical world.

The fact that Jesus was appearing as an angel and not in his physical body, is also evidenced by other angels appearing in the tomb:
Now Mary stood outside the tomb crying. As she wept, she bent over to look into the tomb and saw two angels in white, seated where Jesus' body had been, one at the head and the other at the foot. (John 20:11-12)
These are obviously spiritual companions-assistants of Jesus - and it clearly describes them as "angels."

Did Jesus' physical body die then?

The scriptural evidence is very clear that Jesus' physical body did indeed die on the cross, and his spiritual self left his body:
When he had received the drink, Jesus said, "It is finished." With that, he bowed his head and gave up his spirit. (John 19:30)
And when Jesus had cried out again in a loud voice, he gave up his spirit. (Matt. 27:50)
Jesus called out with a loud voice, "Father, into your hands I commit my spirit." When he had said this, he breathed his last. (Luke 24:46)
With a loud cry, Jesus breathed his last. (Mark 15:37)
These verses clearly describe that Jesus' physical body died, and his spiritual self ("his spirit") left his physical body. To "give up" - translated from the Greek word ἀφίημι (aphiēmi), meaning "to depart" or "to let go" or "to leave" - refers to the separation of the spirit from the physical body.

And "breathed his last" - from the Greek word ἐκπνέω (ekpneō), meaning "to breathe out one's last" and "to expire" - refers to the finality of the physical body's death.

Note that "spirit" in these verses is being translated from the Greek word πνεῦμα (pneuma) which means, according to the lexicon, "the vital principal by which the body is animated" and "the power by which the human being feels, thinks, decides" and "a human soul that has left the body."

These definitions indicate clearly that the "spirit" is the personality that occupies the physical body, and that person can separate from the physical body at the time of death. This is what occurs at the time of death for everyone: Our spirit-person disconnects from the flesh of the physical body.

Though we certainly cannot compare Jesus' physical body with the physical body of others due to Jesus' relationship with the Supreme Being, it is also clear from the texts that Jesus' physical body did certainly die:
Because the Jewish leaders did not want the bodies left on the crosses during the Sabbath, they asked Pilate to have the legs broken and the bodies taken down. (John 19:31)
This clearly states that what was left on the crosses were dead bodies. Bodies absent of the living force - the spiritual self that once resided in each of them.

Wasn't his body put in a temporary tomb?

We can also support the reality that Jesus' body was moved from the tomb by the fact that the initial tomb Jesus was laid in was not his own. It is clear that this tomb was a temporary tomb and not intended for the body's final resting place:
At the place where Jesus was crucified, there was a garden, and in the garden a new tomb, in which no one had ever been laid. Because it was the Jewish day of Preparation and since the tomb was nearby, they laid Jesus there. (John 19:41-42)
The person responsible for moving Jesus' body to a temporary tomb is Joseph of Arimathea. Joseph is described as a secret disciple of Jesus in three of the Gospels, but Joseph was also a Jew and part of the institutional temple Council (likely why he was a secretive disciple):
Later, Joseph of Arimathea asked Pilate for the body of Jesus. Now Joseph was a disciple of Jesus, but secretly because he feared the institutional temple leaders. With Pilate's permission, he came and took the body away. (John 19:38)
And it is clear that Joseph needed to follow Jewish customs of the day:
Joseph of Arimathea, a prominent member of the Council... (Mark 15:43)
Joseph was also a rich man - a man of influence:
As evening approached, there came a rich man from Arimathea, named Joseph, who had himself become a disciple of Jesus. (Matt. 27:57)
And so, as a "prominent member" of the Jewish Council, Joseph needed to follow Jewish custom. The custom of the day was that a person's dead body is placed in a tomb assigned to the family of that person. In other words, it was not proper that Jesus' body be laid in a tomb that did not belong to him or his family. So Joseph was essentially breaking Jewish custom if he had kept Jesus' body in that tomb.

And while Joseph was a man of means, Joseph would not have purchased a permanent tomb for Jesus, as this would have revealed him as Jesus' disciple - information he was guarding. Jewish law strictly regulated burials. Joseph asked Pilate - not the institutional temple priests - for permission to take Jesus' body. This is consistent with Joseph acting in a clandestine manner.

Didn't they fear Jesus' body was going to be moved?

Adding to this is the fact that the institutional temple priests were concerned about Jesus' prediction that he would rise after three days. So there is the potential for his body to have been moved by the institutional temple priests, by Joseph and his people, or by the Romans, to another tomb.

The fact that there was significant concern about Jesus' body being taken from the tomb is confirmed in Matthew:
"Sir," they said, "we remember that while he was still alive that deceiver said, 'After three days I will rise again.' So give the order for the tomb to be made secure until the third day. Otherwise, his disciples may come and steal the body and tell the people that he has been raised from the dead. This last deception will be worse than the first." "Take a guard," Pilate answered. "Go, make the tomb as secure as you know how." So they went and made the tomb secure by putting a seal on the stone and posting the guard. (Matt. 27:62-66)

How did the stone get moved?

In Matthew, the guards were still there when the Marys came, and it describes an earthquake and an angel rolling away the stone.

But in Mark, Mary and James' mother found the stone was already rolled away - and there were no guards posted or around at all.
Very early on the first day of the week, just after sunrise, they were on their way to the tomb and they asked each other, "Who will roll the stone away from the entrance of the tomb?" But when they looked up, they saw that the stone, which was very large, had been rolled away. (Mark 16:2-4)
In the Book of Luke, they also found the stone rolled away, and no mention of any guards:
Early on the first day of the week, while it was still dark, Mary Magdalene went to the tomb and saw that the stone had been removed from the entrance. but when they entered, they did not find the body of the Lord Jesus. (Luke 24:1)
Same with the Book of John. No mention of any guards, or the tomb being guarded:
On the first day of the week, very early in the morning, the women took the spices they had prepared and went to the tomb. They found the stone rolled away from the tomb (John 19:1)
Thus, three of the four Gospels describe them finding the stone already rolled away, with no guards in sight. By virtue of majority, this means that virtually anyone could have rolled away the stone - since it was rolled into place by people, it could have been rolled away by people just as well. And as indicated in Matthew, multiple parties were interested in Jesus' physical body - including the institutional temple priests, the Romans, Joseph, Mary and Jesus' other disciples.

And in the only account that has the guards posted, there is a significant inconsistency elsewhere. The Book of Matthew also portrays additional events regarding the stone that are not portrayed in the other Gospels:
There was a violent earthquake, for an angel of the Lord came down from heaven and, going to the tomb, rolled back the stone and sat on it. (Matt. 28:2)
One would think that this "violent earthquake" would have been significant enough to report in the other Gospels. This earthquake occurs when the two Marys went to the tomb according to Matthew only:
After the Sabbath, at dawn on the first day of the week, Mary Magdalene and the other Mary went to look at the tomb. (Matt. 28:1)
The other three Gospels describe Mary only finding the tomb open and the tomb stone rolled away, with no mention of the earthquake, any guards, or the stone being rolled away by the angel.

This isn't the only contradiction among the four Gospels regarding this event. In fact, all four differ in their portrayal of Jesus' initial appearance:

Why are there three different versions?

In the Book of John, Jesus first appears to Mary after she saw two angels sitting in the tomb.

In Matthew, he first appears to multiple disciples after the angel who rolled away the stone appeared to the two Marys in the tomb (bringing up the question - if Jesus' physical body left the tomb to appear to the disciples, how did the physical body get out of the tomb since the tomb was just opened by the angel with the two Marys watching?). 

In Mark, a "young man" first appears to Mary and James' mother, and then Jesus appears to Mary first.

In Luke Jesus appears to Peter and Cleopas first as they were walking as quoted above.

Why the discrepancy between these Gospel accounts regarding who saw Jesus first as well as how the stone was rolled away?

These inconsistencies make it clear that these descriptions are coming from different sources.

What would you think if four different people came to you to tell you something happened, and each description was different? Who would you believe?

One might prescribe that only one version could be right. This would make the other versions wrong. But they could still each be telling the story as they remembered it. And certainly, in such an angelic appearance, each may have seen different things.

What is consistent among all of these appearances is that Jesus did not appear in his physical body. If he did, they would have immediately recognized him each and every time. These were his closest disciples - close enough to Jesus to be able to immediately recognize his physical body.

And since they didn't, and since "he appeared to them in a different form" we must accept that Jesus' physical body did not "rise from the dead."

Yet Jesus himself indeed did "rise from the dead" because he appeared after the death of his physical body.

And what does his appearance teach us? Why did Jesus do this? To prove that he was great?

Don't be ridiculous.

Jesus reappeared after the death of his physical body to continue his instruction to his students, but also to illustrate that we live on after our physical body dies. We have a spiritual identity that is eternal and doesn't die when the physical body dies. This is the important teaching of this event. The details do not matter. Jesus was trying to show those who had not truly heard his teachings that we are not these physical bodies - evidenced by such statements as:
"Do not be afraid of those who kill the body but cannot kill the soul." (Matt. 10:28)
This doesn't mean Jesus' physical body was like ours. His physical body was also spiritual due to Jesus' relationship with the Supreme Being - and Jesus' use of his body in his service to the Supreme Being.