Mrs. Shipira, I believe the only difference is in the orthography. Skillfully... is the American English spelling. Skilfully... is the UK English spelling.

Denis is correct.

There is no definitive different between "skilfully" and "skillfully". The only main different is the fact that one is the British spelling, and the other is the American spelling. Both of which mean the same thing and can be used the exact same way. Although "Skillfully" is both my preferred way of spelling, and is mostly used by others, there is no issue with spelling it as "skilfully".

A person can be skilled or skillful, but a thing can only be skillful. This is sometimes an important distinction: "skillful work" is work done with skill; "skilled work" is work calling for a skilled (or skillful) person to execute it. Also, only skillful has an adverbial form, so for consistency one may want to say "skillful" if one later intends to say "skillfully."