As an adult, who frequently plays classics on both the NES and MAME, my entire perspective has changed. The games that seemed so amazing as a kid have not aged well, most of them are shallow, repetitive, and cheap, compared to their NES counterparts, and despite the graphical differences, I feel the NES counterparts have largely aged better and are far more satisfying.
Here are the games that I feel are superior to their arcade counterparts, note that not all of these are ports, and for the purpose of this list, I am strictly going on games that have the same name, even if the game was entirely different than the arcade version. I think some of these choices are going to be no-brainers that everyone will agree with, others will be more controversial, let me know your thoughts, and feel free to add in games that you feel match this category.
1) Ninja Gaiden--no contest here, the arcade game is a shallow and terrible beat em up, the nes game is an amazing action platformer thaty revolutionized story telling in video games.
2) Double Dragon 1-3: The first game is controversial, although I prefer the home port of the original more than the arcade version and feel that the more robust move set combined with better level variety make it a much better game, I get that many will disagree with me. I think the next 2 games, there is no contest, NES dd 2 and 3 are far superior to their arcade counterparts in every way with deeper fighting mechanics, more fleshed out stories, and some nice level variety.
3) Bionic Commando--the arcade game takes a cool gimmick of having a bionic arm and ruins it with cheap gameplay, and quarter muching gameplay. The nes version takes the same concept and adds in a cool map, a nice variety of upgrades, extremely well crafted levels designed to make use of the bionic arm, and the greatest ending to any game ever, that has you literally shooting Hitler in the face with a bazooka.
4) Battletoads--While the arcade game is fun, it is basically just another beat em up, sure there is some level variety, some really cool visual moves, and a legit multiplayer component, but the nes game is one of the most unique, challenging, and varied games to ever exist. Unlike many of the game I already listed, arcade battletoads is actually a very solid game, but it pales in comparison to the greatness of its nes counterpart.
5) Strider--I know I am probably the only one who will hold this opinion, the arcade game is great, with some of the coolest set pieces and most bad ass weapon to ever grace the arcade scene, but the gameplay itself has not aged amazingly. While still a great game, I really enjoy the more open ended feel of the nes game to the somewhat cheap action of the arcade.
6) Contra: I never actually played the arcade game as a kid, I only recently tried it on mame, and my god is the arcade game shit. The controls are wonky, the graphics are weird, and there is just something off about the gameplay. In comparison NES contra is one of the greatest games ever made and destroys its arcade counterpart in every way shape and form. I have not played the arcade Super C but I wonder if the same holds true there.
7) Punch out: The arcade game has stiff controls, limited opponents, and it is hard for all the wrong reasons. The nes game takes the same concept and adds in a wealth of new characters, tight controls, loads of personality, and a ton of secrets that can be used against the opponents. There is no comparison here, the NES game just blows the arcade version, and its sequel out of the water.
I'll stop there for now, let me know your thoughts, agree? Disagree? Games you want to add to the list?