The 6xx are harder to drive, so a amp/dac may be needed. The 58x should be easy enough to run on a phone or laptop, but a dedicated amp/dac would be advised to get better sound quality and power.
Build is mostly the same while the 6xx come in a nice foam case and are midnight blue color while the 58x only come in a cardboard box and are gloss black in color.
As for sound summery. The 6xx is smoother, warmer, more mid/upper bass fullness but less sub bass extension. 6xx have slightly more forward vocals. Treble is a bit darker, but overall provides a bit more resolution. Depending on sound preferences you could like either one I prefer the 6xx.