Both are colloquial English terms commonly heard in USA for friend or a term used to address a stranger in a friendly manner . Bro is shorthand for brother and is more commonly used with Caucasians in South Africa and USA whilst homie is short hand for home boy which means friend from your neighbourhood. Homie is more commonly with black people in USA. Neither terms are commonly used in British English or Australian English where we typically say "mate", "dude" or "pal"

I don't think "homie" is used very often anymore. With a lot of these informal expressions, they are not used very long before being replaced by something else. Personally, I would find it strange for someone nowadays to use the word "homie." "Bro," however, is still very commonly used.

There are no objective differences concerning "Slang". Neither are there Rules for the use of "Slang". That is why it is referred to as "Slang" rather than Grammar.