Cost of living in Durham will be cheaper, however, there are pockets that you'd likely want to stay out of. If working in Durham you could easily live as far east as Raleigh if you have a car or go more west past Durham. I live in Raleigh and was born and raised near Burbank. I can tell you that knowing what I know now I'd choose Durham all the way.
Burbank living is similar to Durham in the sense that there are pockets. You likely wouldn't live in Burbank anyway, you'd live somewhere in LA though. Your money won't go as far, is the job paying the same regardless or is it not a paid internship?
Is the job dealing with movie studios? If so that changes everything. You want to be in LA, really. They have Burbank studios, Santa Clarita Studios and a few others then of course your big ones. If all else fails you can move to Van Nuys and do porn.
More info about the internship would be needed for anyone to really help guide you.