Between 2004-2012 when the show was airing, Oxycontin and other pure opioids were classified as Schedule II drugs (presenting "significant risk of harm and dependency"). Vicodin was, by contrast, merely a Schedule III drug (presenting "moderate or low risk of physical dependence")
In practical terms, the major difference is that a Schedule III drug can be repeatedly prescribed by the same doctor without significant oversight, the prescription can be renewed by the patient multiple times without needing additional permission from the prescribing physician (which is perfect for House, because he's taking far too much and regularly stealing Vicodin from his patients) and unlike Schedule II drugs, there's no significant paper-trail and copies of the prescription don't get sent to the DEA.
We see House doing all of the above repeatedly. Prescribing for his patients then stealing their drugs, doctoring scrips from other doctors, taking another doctor's prescription pad, refilling his prescription repeatedly without permission, etc, etc.
The actions listed would be near-impossible if he was taking a pure opioid, especially in New Jersey where Schedule II prescriptions were automatically sent to the State and DEA for audit and investigation.
As to why he doesn't take stronger meds, we do see him occasionally experimenting with other medications like heroin, ketamine, methadone, etc. They harm his ability to think and reason, which, we learn, is one of the things that prevented him from committing suicide after he damaged his leg.
House: I'm done with the methadone. I screwed up.
Cuddy: You're afraid of change. The one thing you have is your intellect. You think if that's compromised, you have nothing. (She holds up the dose of methadone) Just take it.
House: No.
