I feel 活用 has some kind of feeling of "put to the use", as a politician would use his charm in a campaign. The other three seem confusing to me, however. What are the differences?
利用 is a very general way of saying 'to use something', and often the purpose is to benefit the user. For example, you can say 「あの会社のサービスを利用する」 (use the services provided by that company), which implies that you use the services for benefits. You can also say 「彼は私を利用して妻を騙した」, which means he tricked his wife indirectly, 'using' me. So 利用 can be used to replace 'use others for one's own ends' in English.
活用 is pretty much the same with 利用 but 活用 means to use something more effectively in order to achieve one's goal. You can say 「通訳は私の日本語を活用できる仕事だ」 (I can use my Japanese if I become an interpreter), which emphasizes that you can fully use and practically use Japanese in order to perform the best during your work (as an interpreter).
An interesting combination of the two words is 利活用, which is mostly used by government officers.
使用 is probably the most directly way of saying 'to use'. It doesn't often convey other meanings than 'to use something'.
用いる is the same as 使う but it emphasizes that, compared to other choices, the user prefers one choice the most and decides to use it. 「この方法を用いて計算しよう」 (Let's use this method to figure it out) emphasizes that they don't pick other ways to figure out the problem, but they took one method which they are particularly in favor of.
Hope this helps!
I could be wrong, as my Japanese was 90% self-taught (in 三重県), but...
My understanding is that 「使用」 is usually for using something physical (especially a consumable or something that experiences wear) and 「利用」 is for something intangible (like a service or someone else's knowledge) especially to one's (possibly ズルい) advantage. Further, 「活用」has always felt to me like it is closer to "apply" (such as a skill or one's own knowledge) than to "use." ※In that regard, "use...effectively...to achieve one's goal" in Greek Fellows' above answer seems spot-on.
Again, this is to the best of my knowledge, based on observation and "trial-and-error," because any time I ask my wife about subtle nuances in Japanese, she says "ほんなもん解らん！やで、私に聞かんと辞書を引けよ！" (of course, anyone searching/posting on this page already knows how fruitless using the dictionary can be, though...) That said, we should all be really grateful forums like this exist!