The Socs, which stands for the "Socials," are rich kids from the West Side, while the Greasers are poor kids from the East Side. The Greasers are also wilder than the Socs. The Socs commit rich kid crimes like having beer parties and jumping Greasers. On the other hand, Greasers steal things and rob gas stations.
In addition, the Socs fight for different reasons than the Greasers do--the Socs seem to fight out of a sense of boredom or ennui. They wear madras and have fancy cars like Corvairs and Mustangs. They have parties and fight each other for kicks, but it's clear that they will still have cushy lives. The Greasers, on the other hand, fight because they need to stick together and their lives are difficult. Many of them have families in which the parents are dead or abusive, and the Greasers regard each other almost as family members. They fight to blow off steam and because the Socs often jump them.
To me, the major difference between these groups is in how society sees them. They are both pretty similar, with teens who have problems and who often turn to violence and gangs to make themselves feel like they are wanted and cared for.
But the greasers are looked down on by much of society. They are seen as the lowlife kids. Meanwhile, the Socs are much more respected because they have money and important parents. Things that they do do not seem like such a big deal to the community because there is this assumption that they are basically okay where the greasers are basically punks.