They're pretty different. Martha's Vineyard is larger and more diverse than Nantucket, especially in the Summer. As people have said, Martha's Vineyard tends to have a more laid-back feel to Nantucket, but it does depend on where on the Vineyard you are. Each village on the Vineyard has a distinct character. Oak Bluffs has the most action, and has the most "urban" feel. The Tabernacle neighborhood in Oak Bluffs is very iconic to the area, as is the Flying Horses Carousel. Vineyard Haven has a great street fair, and a cute town center on the waterfront and the famous Black Dog Tavern. Edgartown has a very quaint center and is the home of the 4th of July Parade and its famous Yacht Club. Tisbury is fairly sparse, although it contains Vineyard Haven. Aquinnah/Gay Head features gorgeous clay cliffs. The Vineyard gets considerably more crowded in the Summer than Nantucket, but has a very well established system of busses, bike/moped rentals, and taxis. The beaches get VERY crowded, although there some diversity in surf depending on if you're on the sound side or the ocean side. There are ample accommodations around the island between the hotels, B&Bs, campgrounds, rentals etc.
Nantucket is much quieter and more outwardly affluent than the Vineyard. Nantucket Village has a lot of great shops and a lot of local charm. Nantucket Nectars is based there. Outside of the village it gets sparse quickly. There are plenty of places to eat and rent lodgings, but it has a much more relaxed and less party feel from the Vineyard. There is not nearly as much connectivity with transit on Nantucket, due to the Village being the only built up destination. Bike rentals are popular, although the aforementioned cobblestones are murder to ride on (and drive on for that matter). The beaches are varied depending on how far out you go. Jetties is the major beach near the Village. The hamlets of Madaket and Siasconset on opposite ends of the island are insular and very wealthy, but have a lot of character. Madaket offers some of the most beautiful sunsets (rivaling Aqiunnah on the Vineyard). The northeastern end of the island is very undeveloped.
