Re-Mixed Music
Finish a game to unlock the option to choose between "arcade" or "re-mixed" music and sound effects in that game's options menu. It doesn't matter how many times you continue.

CPU Battle Mode
Finish a game without continuing to unlock the "CPU Battle" mode in that game. CPU Battle lets you practice against the CPU opponent of your choosing without diving into the arcade mode.

Deluxe Versus Mode
Finish all three games without continuing to unlock the "Deluxe Versus" mode in the compilation's main menu. Deluxe Versus allows you and a friend to mix and match the characters from various games. For instance, Hyper Fighting Ken against Champion Edition Ryu. You can also crank the speed in this mode up much faster than the default setting in any of the three games.

Unlock 2nd Artwork Galleries
Finish each game to unlock a second artwork gallery for that game in the compilation's "collection" menu.

Best Credits Sequence
No matter how you manage to beat a game, you'll see the same ending sequence for the character you chose. However, what goes on during the credit roll is different depending on how many rounds you lost and how many times you continued. You'll get one sequence if you beat the game, but had to continue to do so. You'll get another, slightly livelier sequence, if you beat it without continuing but lost at least a single round. To get the "best" sequence, finish a game without losing a single round.

Practice Against Bosses in SF2: The World Warrior
You can't play as the bosses in the first SF2 game, but you can at least practice against them. While selecting your opponent in the "training" or "CPU battle" modes, move the cursor to the left of Ryu and Ken, or to the right of Guile and X to set Balrog, Vega, Sagat, or M.Bison as your opponent.