The difference between these is that drug abuse does not necessarily involve drug addiction, but drug addiction necessarily involves drug abuse.
Drug abuse is simply the use of drugs in a way that is not consistent with medical needs or with what society thinks is acceptable. If I go out and binge drink until I pass out, I have abused a drug (alcohol). However, I am not necessarily addicted (I may not be an alcoholic).
Drug addiction is a physical and psychological thing. It is when you cannot do without the drug. If you do not have the drug, you experience symptoms of withdrawal. If you are this dependent on the drug, you have clearly abused it.