There isn't a difference really. They both just mean nonsense. If I was forced to think of one difference, I would say we are more likely to say someone is talking gibberish, than they are talking mumbo jumbo. Mumbo jumbo is used more to describe a thing, rather than the act of talking. But, that said, if you said he's just talking mumbo jumbo, no one is going to tell you it's wrong.

"mumbo jumbo" & "gibberish" mean the same thing. However, "mumbo jumbo" is a spell or incantation used by blacks for magical purposes like dispelling troubled ancestral spirits. Hence gibberish to non-believers.

"Mumbo jumbo" is more informal, almost a slang form of "gibberish." In writing I would use gibberish.