To me, there isn't much difference between use, utilise, and employ. Employ and utilise are a bit more formal and perhaps pretentious than use.
Exploit means to take advantage of, and would refer to doing something immoral or illegal. It has definite negative connotations.
Apply would rarely mean 'use'. I can't think of an example where it does. It means to put something on. I guess the meaning could be similar in some cases, but there is always quite a distinct meaning.