I think in humans burps are a release of swallowed air, while farts are gasses produced by fermentation in the intestines. Ruminants OTOH have a fermentation chamber (rumen) as the first part of their digestive system (their first 'stomach'). Other animals have different fermentation strategies.
Human burps are air that you swallowed, or from CO2 from either drinks or produced when stomach acid mixes with bicarbonate. That CO2 does not necessarily has to come out as a burp, it can also pass into the blood stream. We don't have a rumen like cows do, where bacteria start breaking down the food. Our stomachs are mostly free of bacteria (except for things like helicobacter pylori). All our bacteria-activity is in the large intestine. A lot of the gases are absorbed and exhaled through the lungs, not all of it has to come out through the rectum.
