Classic

• The main attraction from the remastered are the upgraded models, but in the standard FF8 you have a great modding community that has already taken care of that and other problems, there are many total conversion mods that will help you fix the old blocky models, there is even a mod to fix the midi music and change it to the original PSX version. So it's up to you if you prefer a game that is ready from the get go or you like modding your games to your taste, but remember, the remastered is charging you extra for that readiness.
• This one has Chocobo World, a legendary mini-game, designed for the PocketStation (that was released exclusively in Japan), that gives you broken items, in the remastered this mini-game was removed and the exclusive items now have to be obtained via one of the characters abilities, making them harder to farm.
• Both this version and the remastered come with in-game cheats or boosters that you can activate at will, but in the remastered achievements will be deactivated if you use them, in the standard version that won't happen. Imagine someone gives you free food and when you are eating it, that person comes and says it wasn't really free and punch you in the face. You shouldn't be punished if you want to play a single player game your own way.
• Talking about achievements, the remastered comes with fewer of them, if that is good or bad depends on you.
• Even if the previous points were obviously one-sided in favor of the standard version, I believe this one will make you take the final decision: The standard version has DRM, pretty stupid from a game with more than 20 years, the first thing you have to do before you play this game is to make a SquareEnix account, so if you buy the game and their servers are down, you are screwed, you can not play a single player game, and they have refused to fix it since the steam release in 2013, and at this point I don't think they ever will.