People should really sometimes check the product page, since they often lean you towards the product most suitable for you, if you are able to read past the marketing hype. Sennheiser has recently changed their product pages, but you can check the current ones, they are quite fitting.
HD600
"Engineered for absolute clarity, the HD 600 is the professional’s choice for reference playback. An instrument of Sennheiser’s passion for perfect sound, it is globally recognized as the standard for analytical listening in both technical and hi-fi applications—even 20-plus years after its release."
HD650
"A benchmark for total audio immersion, the HD 650 reproduces the subtle details in your tracks with the same confidence as the highlights. With a fluid signature tuned for hours of fatigue-free listening, it is the headphone to reach for when the music matters most."
HD650s were not made to fix anything, they are just ever so slightly more premium product with minor tweaks that result in more relaxed tuning - nothing more, nothing less.
People talk seriously about some Beyerdynamic headphones as hifi cans, when they really exist for audio production, just as their product page suggests. They have likeable qualities for listening music surely and many prefer them for that, but in the end, it is no surprise that after hearing the famous treble peak they get returned back to the store.