I was wondering which of the two to buy. I want a pirate game but I do not know which one to choose.

Which is better?

I haven't played Corsair as yet, so can't tell you about that.

What I can talk about is Man O War, and I will start by summing up that even if it was free I wouldn't recommend it. Now let me drill down as to why:
Graphically it is moderate to poor. Seviceable maybe, bland and empty would be more words for it.
Animation, if you took several new people to animation they would come up with this dreadful stick like movement. None of the fluidity of movement you would get with any other game, its actually laughable in most places, but even worse in others.
Setting, I really like Warhammer world, and would have liked this have some of that. As it is, the "cities" have about 3 npcs in them, one even may walk about!
Quests, probably two or three uniques, go here-kill that, go here-deliver this, sort of thing. Couldn't see any overiding point to it.
Bugs, so many, from your ship stuck just below the water line, to floating ships, or getting teleported from town to the middle of the sea. Obviously pre-alpha release was fine.

So if I was given the choice between this one and any game (other than maybe Tempest), I would take the other game, especially when you see the hilarious £16 price with a reduction (I paid full pre-order). Play anything other than this, I might try Sea Dogs - City of Abandoned Ships next as that gets good reviews. Sid Meiers Pirates is quite fun. AC: Black Sail, and Rogue of course have all the piratey action and other then the endless collectibles and modern day parts, are very enjoyable. So is Risen 2 and 3, which are quite piratey.

I'm afraid I've never played a Man O'War gme, but if you haven't yet, I'd suggest getting Assassin's Creed Black Flag. It's both a great Assassin's Creed game and a great pirate game. In fact, some people argue that if it wasn't for dropping the words Templar and Assassins here and there, it almost wouldn't be an AC game. I kinda disagree with that, but at the same time I'd say it is definitely the one with the least amount of references to the AC lore (except for present day parts which can be done quick if you don't care about them).

Depends on whether you prefer Sea Dogs/PotC/Age of Pirates or Sie Meier's Pirates/Port Royale/etc.

I'm in the first crowd, myself.

I've not played Man o' War, but I have played Tempest. Didn't like it at all, unfortunately, because it's just a bad mobile game port that gets tired and repetitive quite quickly. The ship-to-ship battles can be quite fun, for a little while, but boarding and fighting with a sword is awful, plus there's various other flaws. I could only recommend it as a time sink, if ever it's available for 90% off.

As others have mentioned the Sea Dogs games, I can't help but add my tuppence there too, as I have the original and City of Abandoned Ships. The ideas behind both can't be faulted, but the execution can, because they're riddled with bugs and problems that even the best mods around can't fix, basically, because the core game is so poorly designed. Very frustrating, because while in principal these games ambitiously offer everything most pirate game lovers would want, all the positive aspects are like a diamond buried deep within an enormous pile of turd.

Sadly, I've still not found any pirate themed game that betters Sid Meier's Pirates! from 2004, simply because it gets all the basics right and is fun to play.

I think what makes it even more frustrating, is although it was known generally as Sea Dogs 2, City of Abandoned Ships was actually the fourth game Akella built around the same engine and concept, with only the Pirates of the Caribbean one missing from GOG. The only thing that improved ever so slightly was the graphics, between the 1st game in 2000 and the fourth in 2009. The UI, controls, bugs, broken quests, etc, all stayed the same.

It's almost like after the first Sea Dogs game, they kept banging themselves over the head with the same hammer. Of course, there appears to be a small group of fans who love the game, come what may. Some will say it's an "old school hardcore" RPG type niche game, but I'll just say it's an utter bloody mess and great potential wasted.

I'm one of those that love the series. And there's a fifth game now. To Each His Own. It's not on GOG yet.

But have they made any improvements on that last release, such as the UI?

I can understand why there's love for the game series, of course, but it's disappointing that the developers don't really seem to have taken any strides towards removing or working on the major flaws, over nine years in the first four releases. That's what I found most frustrating.

Windward is pretty fun.

I own both games but I've only played Tempest and in my experience with Tempest is it's a broken piece of shit most of the time. Nothing but issues ever since I've bought it.