While they do mean the same thing, the notion of perceiving time is perhaps slightly different. "I will stay there for more than a year" looks at your year of staying there as a single block of time, and the moments within that year are somehow not considered as important. "I will be staying there for more than a year" emphasizes the moments that make up the year of staying there. It's the same block of time, but perceived a bit differently. You're thinking more about all of the days and months that make up that year.

I agree with Alan. The use of the continuous tense nearly always puts the emphasis on the experience or the activity itself.

It means the same thing :)