"Qu'est-ce tu fous" and "Qu'est-ce que tu fais"
What is the difference between these sentences? And why "fous"? Why not "fais"?
The other difference between your two sentences was 'que' being dropped in the informal/vulgar version. It is rather common.
Colloquially, we try to drop as many mute e's as possible, and with 'Qu'est-ce que tu fais ?' we would have 'kes-ktu-fè', which is too hard to pronounce, so either we keep the 'e' of 'que' (kes-keu-tu-fè) or we drop 'que' altogether (kes-tu-fè), which is very common but grammatically incorrect.
"Fous" from the verb "foutre" is borderline offensive and definitely vulgar. It is used quite often so people tend to forget where it comes from and it loses some impact but "foutre" as a noun means "cum" and the verb apparently meant "to ❤❤❤❤" in Old French.
To give you an English equivalent, it is at least as vulgar as saying: What the ❤❤❤❤ are you doing?
What the heck are you doing? = Qu'est-ce que tu fiches? (basically the same as above but when trying to be more polite).
By the way, "foutre" can also be used to mean "to put":
E.g.: Où est-ce que j'ai foutu mes chaussures.
E.g.: Je me fous de sa gueule.
"to not give a f***"
E.g.:On s'en fout.
And probably some other things that are not going through my mind at the moment.