Is burping the same as throwing up air?

Answer 1

Yes and no.

They are the same in that both involve a material travelling from the stomach back up the esophagus and out the mouth.

Where they differ is that burping is just a passive relaxation of the lower esophageal sphincter to allow trapped gas to escape on its own. Vomiting is an active reflex where muscles are commanded to contract and propel the liquid out.