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The main difference between Super Saiyan and Super Saiyan 2 is the difference in power.
According to the Dragon Ball Super Exciting Guide databook, super saiyan 2 is two times more powerful then a super saiyan, which means:
Super Saiyan 2 = 2xSuper Saiyan
There are other differences between both states, you see a super saiyan 2 has sparks of lighting surrounding his aura and also has a different hairstyle. Also it should be noted that in the anime, TOEI sometimes forgot to add the sparks of lighting to the super saiyan 2 aura, which makes it easier for the the fans who have not read the manga to confuse both states.
Hope it helped.