Oxycodone is basically 14-hydroxy-hydrocodone (it has another oxygen atom inserted on a certain position of the hydrocodone molecule). Oxycodone is generally regarded as more stimulating (less sedating) and much more powerful (milligram for milligram) than hydrocodone. It is in widespread usage for moderate to severe pain in first world countries in various dosage forms (oral pills, oral extended release i.e. Oxycontin, oral liquid, injections)
Hydrocodone is more known in the US and Canada, it is the active ingredient in e.g. Vicodin and Norco. It is regarded as a much weaker opiate than oxycodone that is more useful for cough suppression and moderate pain. It is generally prescribed in combination with acetaminophen or antihistamines in syrups and pills. Some people make said syrups into "sizzurp" by mixing them with fizzy drinks and occasionally hard candy or jelly beans.
All painkillers are formulated differently. There are actually several opiates in common worldwide usage that are compounded with APAP. For instace there are hydrocodone/APAP (Vicodin, Norco), hydrocodone/ibuprofen (Vicoprofen), oxycodone/APAP (Percocet), oxycodone/aspirin (Percodan), codeine/APAP (Tylenol number series, plus caffeine in Canada), dihydrocodeine/APAP (co-dydramol) and many more. In addition there are hundreds of single-ingredient formulations ranging from hydromorphinol to ethylmorphine in various compounding forms. In practical usage, however, there are only about 6 to 8 true or semi-synthetic opiates that are in active usage worldwide.
This doesn't even begin to touch on fully synthetic drugs like tramadol, methadone, fentanyl and such.
