Nassau is not an island. It is a city on New Providence Island.
Atlantis is on Paradise Island, which is next to New Providence, connected by a bridge. YOu can get there via water taxi, car, bus, or walk (a looooooong walk from the dock)
The Dolphin Encounter is located on "Blue Lagoon" Island, reported to be where some of the Gilligan Islands exteriors were shot.
That that our Bahamas geography lesson is over (LOL), I have been to Nassau 11 times, via cruise ship and air. If you have never been and want to see it, there is a nice horse drawn carraige ride through part of the city. The straw market is a large place with many natives selling tee shirts (3 for $5, 4 for $5, or even 5 for $5!), decorated straw purses, bags, coasters, hats the usual tourist stuff.
I don't get off the ship anymore unless it is to buy a couple of bottles of liquor. IF you are only buying liquor to bring home and only want your allowed limit, check the duty free liquor shop on the ship, the prices were about $1.00 cheaper.
I went to Atlantis many years ago when it was the Merv Griffin Hotel and Casino.
If you don't want to spend the $$$ for the Dolphin Encounter, and you don't want to rent a day room for the private beach at Atlantis, and you don't need or want a bunch of straw suveniers, then I say stay on the ship.
Castaway Cay is a billion times better beach experience.