'Dipsomania' isn't, as far as I'm aware, a term which is used within the medical profession. So the only definition is the 'everyday' one which refers to someone who has a continuous craving for alcohol.
'Alcoholism' is, of course, recognised by the medical profession. Many doctors would classify someone as suffering from alcoholism when they reach the point at which withdrawal from alcohol has some noticeable effect on the patient's life. (e.g. anyone who says "I don't have to have a drink every night but I can't sleep properly if I don't" is, within the strict medical definition, suffering from alcoholism.
So, someone who feels the need for a few large whiskies each night before they go to bed could (by medical definitions) be classified as an alcoholic but, clearly, they're not a dipsomaniac. On the other hand, it's difficult to see how anyone could be a dipsomaniac without being an alcoholic - unless, possibly, they're living somewhere where they have no access to alcohol but still have the craving.
So, in general, it would seem that dipsomania is a condition which is encountered by a subset of those people who suffer from alcoholism.
Chris