Street Fighter Alpha 2 is an improvement over its predecessor in every way that matters...
There are 20 characters including some that were in Street Fighter 2 like Dhalism and Zangief which never made it to the first Alpha game. There are still some missing characters from SF2 but they were added on to Alpha 3 and two hidden ones. Even with 20 characters the game is fairly balanced with the exception of Akuma, Shin Akuma and Evil Ryu who are on another level.
One of things that Alpha 2 did was get rid of the chain combo's and replace them with a Custom Cumbo which lets you create your own unique combo's in a short time span that uses up your super bar. The super bar is essentially a bar that fills us when you do attack or take damage. There are three levels of it. When a level is full you can either do a custom cumbo or a special attack like the Shinryuken. The higher its level its filled up the more damage the special attack can do and the longer the custom combo is in effect.
You dont really find out about the characters story until the last battle. There is also an optional mid-boss fight which is triggered under certain conditions. There is dialague between the characters which provides some story. Some of it is hilarious like the Ken and Dan one. The main one story about the game is the Dark Hadou which is the killing portion of the Ansatsuken style which Ryu and Ken use although Ken is unaffected by it. Akuma tells Ryu that he has a dark power within him. I suggest you watch Street Fighter Alpha: The Animated Movie for more backround regarding it but its not canon to the game.
Overall its a great game when it came out and it still is today that shouldn't be missed especially if you are a fan of fighting games.