Not really pirate games per se, but there are two promising naval indie games about whale hunting that I'd like to share.
Nnatucket (Picaresque Studio) is a strategy seafaring game set around the 1820s, heavily inspired by Moby Dick (it can be considered kind of a sequel, as you can play as Ismael if you so choose). You have to hire and manage a crew with different specialisations, upgrade your ship and of course sail around the world in pursuit of whales to fill your hold with precious oil, all while fending off pirates and dealing with different events that pop up during your voyage. It has a tabletop-like combat system for both whale hunting and pirate-fighting, with card decks and dice. It's already in its closed beta phase and will be releasing shortly, one would expect.
This next one, on the other hand is still very early in development, I discovered it today precisely through Picaresque's twitter account.
The Whaler (Trym Studios) has a very different approach. It aims to be a first person simulation of whale hunting in the 17th-18th century with elements of sailing, crafting and trading, with a fully explorable and customizable ship. Some of the planned features:
As I say, still very early in development, they will go on Kickstarter next year.