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!
is an older word.
Old English bealcan “bring up wind from the stomach,” also “swell, heave,” of echoic origin (compare Dutch balken “to bray, shout”). Extended to volcanoes, cannons, etc. 1570s. Related: Belched; belching. As a noun, “an act of belching,” it is recorded from 1510s; also slang for “poor beer, malt liquor” (1706).
Burp is more recent
1932, noun and verb, American English, apparently imitative. The transitive sense of the verb is attested by 1940. Related: Burped; burping. Burp-gun attested from 1945.
An older word from the Latin is probably…
“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.