I was taught that the two mean and are used the same, so they would both be "to stay." The point that I was trying to make is that «stare», «restare», and «rimanere» are synonyms. The only specialty lies with «stare», which can be used for certain states of well-being: e.g. «Sto bene.» or «Come stai?».
Adding another verb and changing stay to a gerund may not make much difference in English, but it is a different sentence. How different in Italian? In conversation I personally think of "will stay" as direct and immediate, and "will be staying" perhaps used when discussing a larger future situation. Example: "I will be staying one night, so there is only so much I can do." Does any part of that context translate?