They are opposites. The HD650 has dim treble, good mids, warmer mid-bass, and more sub-bass. The HD598 has a brighter more forward treble, similar mids, still bright, and much less mid-bass and less sub-bass ability.
The HD650 is not super easy to drive, it needs a competent amplifier. The E10 can handle it, but not to full dynamic range (at least according to some).
As to which is better, neither. They are merely preferences. For your needs, it depends on what you prefer. Bass and dim treble presentation, or a forward treble/mids presentation.
I would suggest some alternatives: Hifiman HE-300 (revision 2), Hifiman HE-400, AudioTechnica AD900, AudioTechnica A900X, AKG Q701, AKG K550, Beyer DT880, Beyer DT770, Denon D2000.