What's the difference? In simple terms, think of Nuevo Vallarta as a suburb of Puerto Vallarta with the airport (PVR) in the middle. Kind of like how the Riviera Maya is a "suburb" of Cancun.
In fact, Nuevo Vallarta is the first in a line of "suburbs" of that makes up Mexico's new Riviera Nayarit. Also included in the Riviera Nayarit are the beach towns of Bucerias, Punta Mita, & Sayulita to name a few.
In Nuevo Vallarta, the hotels & resorts are mostly all-inclusive with multiple restaurants on site. Therefore, guests really have no need to leave the premises. In Puerto Vallarta, most of the hotels pre-date the all-inclusive wave. There is no doubt you're in a city and NOT the 'burbs. If you want a nice dinner, you shower up, put on a clean shirt, and find a nice restaurant every night. Puerto Vallarta has a variety of restaurants (some world-class) for a variety of tastes and budgets. Afterwards, Puerto Vallarta offers a bustling nightlife. There are a multitude of bars & clubs to choose from to party all night if you wish.