The simple answer is severity, but let me expand on this further.
Technically, dysthymia is a pervasive "low level" depression that lasts a long time – often a few years. "Major Depression" is a discrete episode of severe depression. When it is gone, the patient is in "remission,” and feels completely normal.
"Recurrent major depression" comprises discrete periods of major depression that come and go, while "major depression in partial remission" is a severe discrete episode that never completely gets better. How does that feel any differently than dysthymia, you might ask? It doesn't. These terms are descriptions, not different diseases.
The problem with these terms is that the medications are pretty much the same: antidepressants. Ostensibly, the diagnosis is supposed to have predictive value. For example, there are high rates of relapse in partially treated depression, while dysthymia does not usually get worse (since, by definition, it is a pervasive low level depression).
Keep in mind, however, that medication is not the only form of treatment for depression. Talk to your doctor about all of the treatment options available to you.