It depends on what fits the game best. For shooters I don't think one is necessarily better than the other. I personally tend to prefer first-person because it's what I'm accustomed to, though depending on the style of gameplay and/or the aesthetic third-person might be better in some cases. For adventure games and platformers, I think third-person is generally better. And for racing games that let you toggle between "inside the car" and third-person cameras, I pick third-person because it helps me drive better as I have a better sense of where my vehicle is and it gives me a better view of the environment in general (while driving IRL is of course necessarily a first-person exercise, it doesn't feel right in racing games to me because it's not actually me sitting in a car).
Now, if you asked me this 20 years ago, I would have said "first-person" all the way, at least when it comes to shooters and other action games. Back in Gen 5, too many developers simply were unable to get the camera right in third-person games, something that's hard to do in a first-person setup, and the controls were often ungainly. That's why I never cared for games like old-school Tomb Raider or MDK. Things didn't really seem to improve much the following generation, though to be fair I didn't get to play as many games that generation for various reasons. I think the first third-person shooter I ever played and enjoyed and didn't feel like I was fighting the camera or poor controls was Gears of War. That fixed over-the-shoulder camera set in a good position (something pioneered by RE4 two years prior) and more standardized FPS-style controls probably helped a lot.