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!
Best Answer
Belch
is an older word.
-Etymonline
Burp is more recent
An older word from the Latin is probably...
eructation (n.)
"belching," 1530s, from Latin eructationem (nominative eructatio) "a belching forth," noun of action from past participle stem of eructare "to belch forth, vomit," from assimilated form of ex "out" (see ex-) + ructare "to belch," from PIE *reug- "to belch"
As I said in a comment, "belch" is probably related to beer, and its consumption. Most native speakers would relate "belch" to a louder and more vulgar description.