I would stay on Tybee in order to maintain that, "Ahhh... I'm at the beach" feeling. Savannah is a beautiful and charming city, but it's on a river and miles from the beach. Similarly, parking and traffic in the heart of Savannah can be a pain in the neck, especially during peak tourist seasons. Use Savannah as a day trip or two. Stroll up and down River Street and definitely take the ghost tour (there's so much history in Savannah). But in the end, I'd opt to wake up to the sound of the ocean outside my door.
Lastly, even though Savannah and Tybee are both very developed, you don't immediately get that sense with either one of them, which I think is sort of cool.