להואיל = Do a favor. Practically is used only as a very polite and a very literary way to ask somebody to do something, or to describe a favor somebody has done. It is so literary, that I suspect in many case it would be used ironically. להועיל = To help, to be util (תועלת = utility). This word is more used than the first one.

I agree with what had been said, and just to add: תואיל comes many times with בטובך and used many times as an over polite exaggeration request. an example can be in a class when the teacher asks the students: תואילו בטובכם לשבת בבקשה an example for the second toil: I thought it might benefit you: חשבתי שזה יכול להועיל לך