After His resurrection, Jesus appeared to disciple Thomas with wounds on His body: "Put your finger here and see my hands, and bring your hand and put it into my side, and do not be unbelieving, but believe." (Jn 20:27) Now how could a glorified body have wounds? If a man dies by being shot, will he rise with a hole in his chest? If he dies a senile and decrepit old man, will he likewise rise senile and decrepit? (St. Thomas argues that "all will rise again in the youthful age.") A permanent cripple like Stephen Hawking surely will not rise with any disability. Well, that's actually an easy problem to solve. Jesus' mission was not fully completed at resurrection. It remained to prove two things to the world: (1) that He was truly resurrected, and (2) that He was glorified. As regards (1), it was useful that Jesus' post-resurrection but pre-ascension body had enough integrity to live but was not perfect in make-up. He postponed going to heaven until such proofs had been supplied, at which point the physical integrity of His body was fully restored.
Jesus showed His wounds to attest to the truth of the resurrection, but His power to come and go through locked doors, to be seen or not seen at will belonged to the glory of His body. St. Thomas quotes Gregory: "The Lord manifested two wonders, which are mutually contrary according to human reason, when after the Resurrection He showed His body as incorruptible and at the same time palpable." (ST, III, 55, 6, reply 2)
I believe there was a certain inevitable if not 100% truthful trick to the manifestations. It concerns the personal identity of the risen human beings with who they were in this life. St. Thomas insists that it will be the "same" body that rises that also died. "For we cannot call it resurrection unless the soul return to the same body, since resurrection is a second rising, and the same thing rises that falls: wherefore resurrection regards the body which after death falls rather than the soul which after death lives. And consequently if it be not the same body which the soul resumes, it will not be a resurrection, but rather the assuming of a new body." (Supplement, 79, 1) I am not fully sure what he means here; perhaps he thinks that the new body will be made of the same type of matter. But it may be that he thinks it will be very much like our bodies in this life.
I believe this is wrong for two reasons. First, in the state of glory, the soul has complete and perfect control over the body; this is precisely one of the two things that make the body impassible. But if the soul is in this sense supreme, then it is "pure act" to the body's 100% passive power. As a result, the soul's personal identity cannot be affected in any way by the resurrection. The identity between the wayfarer in this life and the person in heaven / paradise is maintained solely through the soul and not in the least through the body, except after the resurrection as the risen saint goes on with his blessed life. For personal identity and continuity of consciousness it is therefore irrelevant whether it will be the "same" body in whatever sense that rises or a different one.
Second, the resurrected body cannot possibly work exactly as a normal body would. Now Adam's body was normal but maintained artificially by unique divine graces. A glorified body is different from both pre- and post-original sin bodies in that it suffices itself. Even if some of its amazing powers, such as "impassibility, subtlety, agility, and clarity," are due to "the dominion of the glorified soul... over the body," still the functioning of the body in paradise will have to be vastly different from our exceedingly complex and weird anatomy and biochemistry that are contingent on the specific design of this world.
(E.g., of what use will our present immune system be in paradise? The über-complex process of blood clotting? The colony of good bacteria in the gut?)
I even entertained the idea that each saint will simply create his own body as he pleases in the state of glory. (This would safeguard maximum freedom of the individual in the communion of saints.) However, there are three decisive objections to this. First, if people have complete freedom to make their own bodies, what's to prevent them from trying on for size crazy bodies? Perhaps in order to build a certain complex machine in paradise, it would be most efficient if I assumed a lizard's body to make work easier. But this seems grotesque. There have to be people in paradise, not hideous chimeric monstrosities. Second, if merit determines the beauty of the body, then complete freedom seems to take away the proper gradations of glory of the body. If, on the contrary, there is no such freedom and one's body is given to him once and for all, then we have the fitting incentive to lead holy active lives here. (This is still consistent with my idea that people can switch between disembodied heaven and embodied paradise at will.) Third, this comes dangerously close to the idea that the body is a mere tool united to the soul accidentally as a hammer to the hand, rather than as an essential aspect of humanity.
But even upon abandoning this hypothesis, it seems that it would have been sufficient for a successful redemption for Jesus to have some human body, as this would adequately indicate that He consented fully to keep human nature, both soul and body, united to Godhead after finding us worthy of His love despite our crimes.
Consider further that a man whose body was burned and whose ashes were scattered, or who was eaten by wild beasts, cannot be resurrected as Jesus was, whose body happened to be relatively intact. If the circumstances had been different, and, say, Jesus had been cut into several pieces, each of which was buried separately, then no resurrection similar to what we find in the Gospels would have been possible. That Jesus had the "same" body was a contingent and "lucky" fact. Why then was it necessary for Him to have it?
I think for two reasons. First, so that He could be recognized physically. For example, "he appeared in another form to two of them walking along on their way to the country" (Mk 16:12), but that might not have worked everywhere. Perhaps having a different even if "better" face, etc. would have hindered Jesus from proving that He lived.
Second, in order to over-demonstrate and make abundantly clear the continuity of personal identity after general resurrection, whenever that will be, since we know that the body is so important for it in this life. Mental illness or dementia or brain damage can destroy one's sense of identity. A senile person may not even know who he is. Jesus suggested forcefully that we will be self-same in heaven as we are here.
There was nothing special, however, about the way Jesus' post-resurrection body looked otherwise.