Some people will claim that nonetheless is used when talking about an amount of something - but I side with those (highly knowledgeable, imho) commenters disagreeing with that assertion on ELU.
Putting that aside, but nevertheless is effectively tautologous, in that either word could be discarded without affecting the meaning at all (but nevertheless, that particular two-word collocation is perfectly common and "natural").
In short, it makes no real difference whether you use nonetheless or nevertheless, OR whether you precede your chosen form by but. In the context of OP's example, all such permutations are precisely equivalent to Although he's busy, he still calls.
It's probably worth explicitly pointing that although the three-word form none the less does exist, and has a somewhat different sense1 to the single-word form, there's no equivalent three-word form never the less in "normal" English. Also note that nonetheless has only recently risen to prominence (and it's still much less common than nevertheless, so my advice would be to stick with the latter).
1 From comments under the ELU question linked to by my first sentence above...
Jim (citing example attempting to prove that nevertheless and nonetheless are different)
Julius II loved him nonetheless for it... means Julius didn't love him any less because of it
Colin Fine (shooting that example down)
Ah. I see the problem. That is not the word nonetheless: it is the phrase none the less. To see the difference, consider Julius did not love him any the less. There is no word anytheless.