Hydrocodone
Hydrocodone is a semi-synthetic opioid which is actually derived from codeine. It is taken orally, and is sometimes used in cough medicine to help people suffering from the flu. It will have a combination of acetaminophen or ibuprofen, and is used quite a bit in the United States. It is typically used for severe pain, and has some addicting side effects.
Oxymorphone
Oxymorphone is also a semi-synthetic opioid analgesic, first developed 100 years ago. It is administered as an injection, or it can be taken orally. It is a preop medication that is used to eliminate apprehension, and is also used to control severe pain. It is also used in veterinary medicine because of its ability to help tranquilize larger animals that need to have veterinarian work done.
Between the two, hydrocodone seems to be the one that is administered for humans, for the most part, in order to control chronic pain. You might want to talk to your doctor about this if you have chronic pain issues, as well as Oxycodone HCL to get their opinion on what might be best for you. Always remember that an addiction to pain meds is always a real probability simply because of how these medications interact with the receptors in your brain. By using these as recommended, addiction should be minimal, and you should be able to get the pain relief that you deserve.