The Ending Explained
Well after just watching the movie (including the ending several times) and reading all the posts on the board, I think I have a pretty good idea of what happened at the ending of the movie. Now, without going into too much complexity about time travel and stuff, several things that you get from the movie help explain what happens. Or you can just jump to "What Exactly Happened"
1) Playing the piano and traveling back and forth isn't random, has to be linked to something between the two settings (1979 and 1999) because we know when Rain plays she arrives on the corresponding date in the school semester for both time settings.
2) When you use time travel to go into the future (1979 ---> 1999) you can only be seen by the first person you yourself see. (examples are like when shes talking at the piano competition but Jay Chou doesn't see here, etc.) HOWEVER, when you go back you must be able to be seen by everybody, because Rain leads a normal life once she arrives back in her time (1999 ---> 1979).
3) Finally, the photo at the end of the movie tells us that time has definitely changed... as in Jay Chou going back has changed what happened to Rain and thus their lives/ world is different from the setting of the movie. We know this because if you read the Chinese on the photo it says it is the 1979 graduation picture (notice same outfits/pins as the day Rain dies in the earlier scene), but now Jay Chou is there and Rain is happy, different hair style etc...
So what exactly happened?
I think that when Jay Chou plays on the piano at the end either the fact that:
(1) he is playing "The Secret" for the very first time, thus sending him to the beginning of the semester in 1979 (like Rain did earlier in meeting Jay when he arrived on the first day) or
(2) the time travel only works for the school semester, and as it has ended when Jay plays, the time travel starts over with first day of 1979 semester again.
But either way Jay arrives at the VERY BEGINNING of the 1979 semester... meeting Rain when she does not know him and has not found "The Secret" yet. (Her smile and then looking down means she doesnt know him yet)
Having decided to stay for good in that time and with everybody able to see him Jay Chou doesn't return at all, Rain no longer finds the secret, no longer dies from an asthma attack, and Jay continues his life from 1979 onwards with her... thus causing their different appearances in the new 1979 timeline with the photograph and a happy ending.
(Probably has to hide his identity from his dad, create a new one, etc.. but it should work out haha)
Whew and that's what I think occurred, any comments/questions welcome.