What's the difference between xuéxí 学习 and xué 学? I understand they both translate to meaning, "to learn, to study" but what is the purpose of this distinction?
There are still a lot of differences in between I think.  


For most cases, when you want a verb meaning "to learn", either of them can be used. Only xuexi is having more syllables to make you sounds more formal and polite.  
When used as a noun, xuexi is more frequently to be used to express "learning".
Also, xue has more meanings rather than "to learn" as a verb. It also means "to imitate", "to perform as"-e.g. shuoxuedouchang（说学逗唱）