I bought an AKG K340 in 1985. I used that headphone for 20 years before discovering Head-Fi. I drove the K340 with whatever headphone output my current gear had, and never really gave it a second thought. The moment I drove that headphone through a good, dedicated headphone amp (Dynalo) I was shocked at what I'd been missing. Successive amps improved even more on that.
What I mean by 'open up' is just that. The soundstage opens up, the imaging improves, bass usually improves and overall balance just gets better. Some headphones sound fine with most amps, but can drastically change character when driven by a high powered, quality amp. The K340, K1000 and HD6XX headphones are three that instantly come to mind. The extra power doesn't just drive them louder, it de-congests them for lack of a better term.
Once properly dampened, most Orthodynamics became far less efficient to drive.