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Arcade games were the gold standard of video games for over two decades. Dedicated arcade machines could have more sophisticated hardware than home consoles and computers were capable of having without being prohibitively expensive. This situation lasted until the mid-90s, when consoles like the PS1 and N64, as well as PCs, caught up and even surpassed arcade machines. Before then, the ultimate complement that could be paid to a video game was "arcade perfect." The arcades were often proving grounds for new home console technology. Sega used its arcade games in this manner, and Namco's games had hardware based on the PlayStation architecture.

For the first few generations of consoles, the manufacturers touted "bringing home the arcade", and Atari used its clout and wealth to be the official home licensee of almost every Japanese game company. Since the 2600, Intellivision, and Odyssey 2 were hopelessly outmatched by arcade technology, they settled for making approximations of arcade games, stripping out as much graphics, sound, and even gameplay as necessary to make them fit on the limited home architecture. But we were happy with our 2600 Pac-Man with the "BAMP-BAMP" eating sound and Donkey Kong with only two levels.that featured an angry gingerbread man and a Mario wearing a red onesie. The Bally Astrocade did a little better, its best-loved game being a pretty faithful home port of The Wizard of Wor.

When the NES came out, it was still not up to par with arcade technology. However, a lot of companies took a different approach. Knowing that their arcade games wouldn't fit in the NES's limitations, they instead decided to work around them by adding content to the home versions of their arcade games.

In some cases, this extra content actually ended up making the NES versions better than the arcade versions to the point where most people don't know a lot of NES games started out in the arcades.

Punch-Out!! The arcade Punch-Out was actually a groundbreaking game in its time. Boxing games until then were either overhead view or side view. Punch-Out gave you an over-the-shoulder perspective, facilitated by the player boxer being a see-through wire-frame figure. This technology wasn't feasible on the NES, so Nintendo worked on the game in other ways, adding a bit of backstory and personality, as well as a lot of new boxers. The wire-frame boxer was not doable on the NES, so Little Mac, a diminutive boxer who was short enough for you to see your opponent, was created. A lot of the personality of the series came from the NES game. Furthermore, the NES game controls far better than the stiff, slow controls of the arcade game, which is available on Switch if you're curious. Few people remember the arcade game, but millions remember Little Mac.

Bionic Commando: The NES Bionic Commando is one of the NES's best-love classics, an adventure with 12 huge levels. The arcade original was a short, weird piece of crap with only four levels.

Rolling Thunder: You might have played the home version, which was released by Tengen on one of its black cartridges since it wasn't licensed by Nintendo. The actual game on the cartridge was in fact made by Namco and was a legitimate Famicom game in Japan. It had more levels and cut-scenes than the arcade game did. 

Ninja Gaiden: The arcade game was an average beat-em-up, the NES game was a side-scrolling adventure with good cut scenes for a NES game.

Double Dragon:  Yes and no on this one. Technos Japan couldn't figure out how to make two simultaneous players work on the NES, so it's a single player game. They did try to make up for it by adding a couple of extra areas and enemies, and they did keep the Billy vs. Jimmy fight (in the arcades, if you beat Willie with two players, your players would then have to duke it out for Marian's affections.) They also added a two player versus fighting mode where you could pick any of the enemy characters except Willy. You could even be Abobo.

TMNT 2: The Arcade Game: This was a home port of the first TMNT arcade game, as it was called TMNT 2 on the NES because Konami had already released a TMNT side-scroller on the NES, Konami was able to figure out two-player simultaneous play, but not four players. This game got several new levels and bosses that didn't appear in the arcade game, including a snow level and a battle with the mutant fly version of Baxter Stockman (the arcade game only featured Baxter in his human form in a flying machine.)

So here are five games that were better on the NES to the point where they eclipsed the arcade originals. Are there any others you can think of?