Utilize = you use a tool employ = you employ someone to work in your company. Then he is employed and you are the employee. So employ = you give someone work exploit = Parasites exploit their host. When countries go into a country and have it as a colony, they exploit the people. Exploiting someone means you take advantage of them, harm them and they gain nothing. apply = you apply something ON something, for example a cream on your skin or a mask on your face. To apply can also mean to write an application for a job though.

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.