I'm a huge fan of the original game, but I'm not aware of any problems with the remaster that mean you should play the original. I think they re-recorded some of the music and I'm not sure if there's an option for the original version, but that's mostly nostalgia for me and not because I consider the re-recorded parts objectively inferior.
The remaster has more control options. Personally I would always use the original tank controls so that doesn't make a difference to me. The original supports tank controls (character relative) and "normal" controls (camera relative). The remaster adds point and click controls.
ConsulCaesar: You can play the remaster with old or new looks, and with old or new controls, your choice. In my opinion both are greatly improved in the new version, but it's up to each player to decide.
No option for old music AFAIK, which is the only reason I would play the original again as it is one of the titans of my childhood.
Elmofongo: Ironically I intend to play it via the tank controls, with a controller especially, since I grew up playing older Resident Evil games.
You have my respect!