I haven't played much in the way of games that compete with Zelda in the gameplay department (i.e. that are like Zelda). Okami is a good example of one, though, and I think I'd rather play it than most Zeldas. Of course, part of the reason why is because I love the atmosphere and narrative a heck of a lot more than those of any Zelda game. It also can more comfortably incorporate anime-esque elements better than Zelda, and I enjoyed that aspect of it. Having played both it and Twilight Princess, I chose Okami as the superior game.
TP might be superior in some ways, but they're not enough. Even the fact that Okami's "dungeons" are crap compared to those of most Zelda games doesn't change my preference. Every part of the game is just so damn fun to play, whereas I can't even get close to saying that about
TP. (I'm comparing these two games specifically because they were out at around the same time and were always compared to each other at the time.)
As far as games I'd rather play are concerned, where those games are
not like Zelda, I'd say my top picks are anything from the Elder Scrolls series, assuming I can get past the incredible difficulty of the intro dungeons of Arena (TES I) and Daggerfall (TES II). Actually I guess I have to qualify this by saying that I'd rather play
any TES game over
most Zeldas, but there's a time and place for each of the two gameplay styles, and I've found myself bored with each at different times. Right now I'm hungering more for TES-style RPG gameplay, so I might feel differently about what I've said here later.
The Dead Space and The Witcher series are also favorites of mine. In fact, I'm uploading gameplay videos of of DS to YouTube, and I'll be doing the same for its sequel and for at least the first The Witcher game. Love 'em.