[Inziladun]

As long as the One Ring endured, Sauron's power that had been contained within it remained intact. Thus Sauron, though substantially weaker than he would have been with the Ring on his finger (in that case the power being fully "his" and at his immediate disposal), continued his existence in a corporal form. The power was still there, he just didn't have access to it.

Though still technically "existing" after the One's destruction, Sauron was reduced to literally a "spirit that gnaws itself in the shadows", to quote Gandalf. He was permanently bereft of such a large portion of his native power that he was incapable of any meaningful action affecting the physical world.

Tolkien said in one of the Letters that if, say Gandalf, had claimed the Ring and had been able to keep it from Sauron in a one-on-one contest, the effect on Sauron would have been identical to that in the event of the Ring's demise: it and the power it contained was lost to him forever.
The permanence of the power, or the lack thereof, is I think, the key here.