The exact reason why she ended their friendship was never super clear . If I remember the book correctly , the big part was because he started associating with his death eaters friends who were actually blood purists and did bad things . Snape tried to apologize for calling her a mudblood but Lily never did restart their friendship leading me to believe it was more because he was friends with all the blood purists and was way into dark magic rather than because he actually believed in all the blood purist stuff . Not to mention in the last book , he gets mad at one of the portraits for saying the word mudblood although that may have just been because of bad memories he has associated with the word .