"SS3 is the level that surpasses the level that surpassed SS1" so says Piccolo. There is a huge difference between SS2 and SS3 the power is greatly increased the hair is a lot longer it goes past the waist, there is more lightning other levels, they look evil and menacing. The transformation from SS2 to SS3 is very big and hard to do it can only be reached through training unlike the other levels. Only Goku and Gotenks have reached this level but Vegeta has surpassed it to SS4 but only with the help of the brute ray.
The advantages as usual are being stronger all round. The disadvantages are that it uses up your energy quickly and when out of energy you turn back to normal SS.