For the purposes of this poll, I would go with Super Mario World. I remember that game amazing me when I played it, and it continually came up with new mechanics, power-ups and worlds to keep the game fresh. I know 3 did the same thing, really, but Super Mario World just perfected it. At least to me.
But the secret best Mario game is Super Mario Bros. 2. I just wanted that on the record.
Super Mario Galaxy is the real answer.
I love them both an awful lot! But 3 is dearer to my heart. I think it was my first video game I ever really played, actually. I can beat SMW in 14 minutes, so that's something.
EDIT: Oh man, you people. I like Yoshi's Island a lot more than Super Mario World. I made a top ten list a while ago of SNES games and SMW didn't even make it on there.
I have the fonder memories for Mario 3, but I believe that Mario World is the better game. The levels were not as bite-sized. There was more reward to exploring the levels because of hidden paths and secrets. There was more variety to the level design. I preferred the difficulty curve of World to Three. The animation and art style is vastly superior. Yoshi. There is a larger variety of enemies to learn about. I prefer the music on a SNES to the music output by an NES.