I've had both of them and of the two the HD600 remained in my stable. "Why?" you may ask an the answer is quite simple. The HD600 were/had: neutral, transparent, detailed, more treble, less mirdrange coloration, much tighter bass, and wider soundstage. They were just perfect for Jazz and older rock. Heck even modern music sounds really good on them. They have just enough treble/bass to not sound tilted and the midrange was not as colored and thus became more revealing.
The HD650 to me sounded too mushy or fluid. They also had a distinct lack of treble and definition of the sound to my ears. They could sound really good on overly bright (aka Hot) mastered recordings but on older stuff that is already smooth they just tended to sound like poo. Also the added bass tended to drown out micro details and also bled into the midrange. The soundstage also sounded quite smaller from the added bass. Any easy way to describe the HD650 sound would be like plugging a 32Ohm headphone into a OTL tube amp with 60Ohms of output impedance. The sounded is full bodied but the bass can become too powerful and muddy , the highs roll off, the mids fill in a bit too much. so to me the HD650 sound represents "tube bloat" that you hear when plugging a lower impedance headphone into a OTL tube amp.
HD600 Sound: Far more neutral, more detailed, more transparent, bigger soundstage.
HD650 Sound: Warmer tube like sound, full bodied bass, rolled highs, compressed soundstage.