Drug abuse and drug addiction are often used interchangeably; however, they are not the same thing. They are closely related to one another, and they do often exist on the same continuum. Drug abuse is the unnecessary or improper use of a chemical substance for non medical purposes. Taking too much of your prescription medicine doesn't necessarily fit that definition. That would be drug misuse. A person that is abusing drugs is using the drug to intentionally "get high." One of the key things to note about drug abuse is that individuals with a substance abuse problem can learn from negative consequences, and they can change their behavior. On the other hand, drug addiction is a physical and/or mental dependence on the drug. The experience of being on the drug is of paramount importance, and all other aspects of life take a backseat to acquiring and using the drug. This focus on the drug generally worsens the person's quality of life, and he/she is almost always unable to change their behavior without outside intervention.

A person with addiction often makes tragic decisions that worsen their situation. . . They often miss work or school, get into legal trouble, endanger their families physically or financially, have terrible health issues, and other serious problems because of their substance use. Even with these dire consequences, people with an addiction cannot change their habits to improve their situation (Bradford Health).

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.