Heh, I was about to go sleep when I realized I had missed the most obvious game of all:
Super Mario Bros - The predecessor to this game was Mario Bros. You know, plain jane, hit the turtle/crab/firefly from underneath Mario Bros. It was just one board at a time with no scrolling. Then SMB came along with 32 side scrolling stages, that took you through this trippy, magical fantasy world. You could get permanent power ups. No game ever did that before. It had permanent music, very catchy. No game had done that before either. It has secrets everywhere including warp zones where you could skip tons of stages at a time. Super Mario Bros wasn't just a huge revolutionary leap over Mario Bros, but it really was a huge revolutionary leap over every video game that had come before it. The biggest leap in all of gaming came from Jumpman himself.
Honorable mention goes to Super Mario Bros 3. Somehow they took the most popular game ever made and then made it a thousand times better. SMB3 is an amazing game, but as far as "most improved" goes, the hands down winner is the first Super Mario game.