Thanks for your reply, Tony and Fox!
The reason that I want to distinguish between these two religious words is that they are concepts that are very important in Buddhism, and there are many many tales about them in our culture. Yet I'm really confused about their English expressions.
From what you have explained to me, may I give two examples to see whether I have grasped your meanings?
1. A woman is killed and her hatred for the killer is so strong that her soul doesn't go to heaven nor to hell, but remains in mortal world. She finds a person and "borrows" her/his body and takes revenges on her killer.
------------ transmigration
2. A woman does many wrongs in her life. When she dies and is put on trial before Hades, Hades says: "You lived a immoral life. As a punishment, you will live as a pig in the other life."
-------------reincarnation
And, could you tell me whether you're disagreeing or not? Your words are too complicated for me.

If I have a clue, it may help me better understand you. Thanks!