In my opinion, the difference between a "drinking problem" and "alcoholism" is that people with drinking problems often haven't yet realized they are--or haven't yet become--an alcoholic.
This is not the same as saying there is no difference, or that every person with a drinking problem is an alcoholic. That's clearly not the case. Some people go through periods of problem drinking caused by a specific event -- grief, loss of a job, being young, going to college -- but then get their act together.
But I would think that if you've tried repeatedly to control your drinking problem and failed -- it's probably not just a drinking problem.
My two cents, anyway.