No, it IS the same "Mystic Pizza", as in the small restaurant. But, you're also right in that the movie is set more in Stonington, and the pizzeria is on the Mystic side of the river. I grew up in the area, I love watching the movie, just for the scenery. Don't expect to get amazing pizza from that restaurant, though--it's never been particularly memorable.
That said, Mystic can be fun to poke around in for a weekend. I particularly like it in the shoulder months--before and after high tourist season. The ocean might be too cold to swim in, but you can enjoy walking on the beaches, shopping, great seafood restaurants, etc.