뭐 is short for 무엇. 무엇 means "what" and 뭐 means "what". This is just like how "can't" and "can not" mean the same thing. It's simply a contraction. The difference between 뭐 and 뭘 that we can see is "ㄹ". 뭘 is contracted from 뭐를/무엇을. So the difference between 뭐 and 뭘 is that the latter has the added object particle "을/를". So in order to understand the difference between the two, you just need to understand the function and use of that particle. 뭐 - "what" with any function in the sentence 뭘 - "what" as the object of the verb in the sentence

Those are the same meaning. You can use both 뭐 and 뭘 to the expression about asking "What?." "뭐" is short for "무엇," and "뭘" is short for "무엇을."