BenKii: The game that comes to mind for me is Metroid 2 for the Gameboy.
The game just isn't very fun on the original B/W Gameboy. The question for me is not "should you play the original?" but more like "which remake to play?" because there are 2 remakes for this game. A 2D unofficial remake for PC called AMR2 and a 2.5D official remake for the Nintendo 3DS called Metroid 2: Samus Returns. While I like both a lot you'll probably have an easier time buying the official version on the 3DS over finding a download for the unofficial remake since Nintendo did a DMCA takedown on all downloads of AMR2. Sucks because it truly is a very good game.
There is, however, one unique thing the original game did that isn't present in either remake.
As you may know, video game music typically repeats after a certain amount of time. Also, older video game music was not stored as a recording, but rather was produced on-the-fly. In order to make polyphonic sounds (in other words, in order to play multiple notes at once), the music was split into separate tracks, each of which plays on a specific sound channel (the number available depended on the hardware). Normally, every track would have the same period, starting over at the same time, so that the music would stay in sync.
In Metroid 2, later in the game the different tracks *don't* have the same period. As a result, one track would reach the end and start over, while the other tracks have not yet reached that point. As a result, the music goes out of sync, and ends up not being exactly the same as when you entered the area. This creates a rather unsettling feeling with the music, and is a technique I've never heard of being done before or since.
The two remakes, like nearly all modern games, just use recorded music, and as a result, are not able to do tricks like this.