Although most dictionaries I have looked these verbs up in define them in very much the same way,
to belch: emit wind noisily from the stomach through the mouth
to burp: noisily release air from the stomach through the mouth; belch
(Oxford Online Dictionary)
I feel (from the times I have come across these verbs in reading) that there is a difference between these two verbs. Can a native speaker of English confirm or infirm, and, of course, should they confirm, refine these definitions, although there is nothing refined about it all!