My personal interpretation is almost the opposite of RH-uems, unfortunately.
To me, "exhilarating" has a much stronger connotation of happiness, while "exciting" is more about adrenaline, heart rate, and so on.
A roller coaster is exciting.
The news that your daughter is going to get married to a wonderful man is exhilarating.
Also, "exhilarating" is much more formal/written/rare than "exciting". For most common situations, use "exciting"--save "exhilarating" for the stuff that's truly special.
"Exciting/ed" can have a second meaning--it can also be about an emotion you expect to have in the FUTURE, but don't have right NOW (this is not always the case--it is only one possible way of using "exciting")
EX: I am bored right now, but I am excited about the vacation I will have later this month.
It would be strange to use "exhilarated" in this sentence. "Exhilarated" will describe the way you feel NOW, not in the future.
(Remember that "-ed" is used to describe the way someone feels himself or herself, while "-ing" is used to describe something or someone that makes other people feel that way.)