I only own the HD 600 not the HD 650/6XX, but fwiw recent measurements from reliable sources show less difference between the two models than apparently existed in the past. The current difference is more like unit-to-unit variation and is less than the difference between fresh earpads and broken-in earpads. In the past the accepted take was that compared to the 600 the 650 had slightly better-extended bass but with what used to be called a veiled treble, meaning rolled off.
Technically speaking, both models have a slight boost in the low mid-range where the bass and baritone male voice lives, but many people actually prefer this over a technical flat tuning.