Bioshock was an amazingly atmospheric game, dark and claustrophobic and twisted and wonderful. Combined with a strong storyline, and solid, inventive gameplay, it's no wonder it attracted a big fanbase. I'd say Bioshock: Infinite surpasses the original in terms of atmosphere, has an equally compelling storyline, and I found I enjoyed the gameplay a little more (there was a lot of changing up of what you needed to do, and whizzing around on those skylines was a huge amount of fun).
On the negative side, this is most definitely a story on rails. Once or twice you have an impression that you might be making significant choices, but this is a skyline with only one destination (and, sadly, one which I found a touch on the predictable side both concerning Elizabeth's nature, and Booker's future). Still, the story, which revolves around a Pinkerton detective named Booker DeWitt sent to retrieve a powerful girl named Elizabeth is very engrossing.
The game world itself it uncomfortable to experience. A glorious city in the sky - full of over-the-top patriotism, strict gender roles, racism, indoctrination and a leader who happily positions himself as god of his own realm. Aspects of that worldbuilding, and the lack of choice in the storyline - particularly some of the people you end up fighting against and slaughtering - makes for some uncomfortable gameplay. Bad enough to be killing off every policeperson in the repressive state, but there's also little choice in who else you end up killing (I say policeperson because, oddly, despite being a strongly patriarchal city, there were a lot of female soldiers).
Elizabeth is an interesting conundrum. She's a young girl, full of book learning, who has been kept in a tower and subjected to cruel experiments. The story is the classic "older cynical man 'shepherds' naive young woman to face tough realities and the question of whether she should kill". When you first find her, the game goes out of its way to reassure you that you don't have to worry about Elizabeth during combat - "she can take care of herself". By this, the game means she can hide behind the nearest solid object, make eeping noises, and occasionally toss you ammo or materialise items. The only person she attempts to fight is Booker himself, and she's more than effective dealing with him on both occasions that she needs to.
Of course, this is a gameplay choice - making the player the active one - but I think I would have rated the game even higher if Elizabeth picked up a gun and shot things as well and we got to switch back and forth between playing Booker and playing Elizabeth. I didn't think much of her shying away from killing while Booker slaughtered thousands while rescuing her.
I preferred the original Bioshock's Little Sisters, who, while still being rescued (or not), gave the impression they would defend themselves against more than the player character. Still, Bioshock: Infinite is several steps up from a game like Ico, where you literally had to lead the girl around by the hand, and she would curl up into a ball if you let her go. (Ico is another beautifully atmospheric game with a strong story, but questionable girl action.) B: I is well recommended, but also made me glad for games like Dragon Age, and doubly happy that the upcoming Torment game is going to let you play the main character as female.