There's been some confusion over the two names Aeris and Aerith. Most people think of the 'Flower Girl' as Aeris, while a few remaining old-timers persist to call her Aerith. Why!? How come!?
The answer's pretty simple. Aerith was the name of the 'Flower Girl' in the Japanese version of Final Fantasy VII. The lucky people who imported Final Fantasy VII before it was translated have stuck to that name ever since. When FFVII came to English shores, the name Aerith was changed to the more smoother Aeris. Hence, that is why the 'Flower Girl' is now popularly known as Aeris Gainsborough.
Apparently there is even yet ANOTHER variation of the name 'Aeris' in FF7. The Japanese programmers behind the game originally called her 'Earith'. As I peeked into the files of Final Fantasy VII (PlayStation), I discovered the game internally refered to Aeris as Earith. However, it would be fair to assume Earith is the rough equivalent of Aerith.
Update (September 2000): J. Piersa has provided some insight into the Japanese language. The 'th' is pronounced more like an 's', and hence Aeris and Aerith sound quite similar in Japanese. It is fairly popular in Japan to say 'thank you' in English, but it'll often come across as 'sank you'. Thanks for the info, Mr. Piersa.