Alcholism is not an illness. Cancer is an illness. Those who want to rid themselves of alcholism enough, do. You cannot willpower an illness away.
Addiction is an illness. It can be an addiction to sugar, booze or drugs, but it's extremely difficult to cure if you have an addictive personality.
Amy Winehouse is a case in point. After she'd been to rehab, she was seen buying and eating vast quantities of sweets, but drink killed her in the end, so it looks as if she needed some kind of addiction otherwise she couldn't get through her day or her life.
Gazza seems to have no support from his family, and if as others have said, they take him to the pub when he's with them, he doesn't need them either. He's had countless courses of rehab, but in the end nothing seems to benefit him.