From what I can tell, they seem to be pretty interchangable, you can do both for a visa, passport, etc. Just wondering whether you can 办 something but not 申请 it, or vice versa?
IMO 办 is a general statement for "deal with / ask people to handle some license related stuff", which includes 申请. So "申请" (apply for) is a subset of "办", whereas you can also say "办手续" but here 办 doesn't mean 申请. Also "办" is more colloquial, and 申请 is more formal. The formal version of 办 is 办理. Additionally, 办 is only for license (usually government related) stuff, but 申请 can be "apply for" for anything. i.e. you can say "申请大学" (apply for college) but you can't say 办大学.