Sennheiser HD6XX series are very different sound signature with AKG series.
HD600 & HD650 are both rich and thick sound headphones with very punchy bass and large in piece. The bass quantity is more than AKG and beyerdynamic. They are one of the best headphones with good details at lower mids and bass at this price range.

HD600 and HD650 are at the same range for sound quality but different sound signature.
HD650 has more bass and darker treble, little empty in mids compare to HD600. It's suitable for rock and other genre that need bass savvy headphones.
HD600 has a fuller and meaty mids, better expansion in treble, which makes it a all rounded headphone to me. It has very good performance for vocal and orchestra.
Sennheiser headphones have good dynamic, which make them excel in big production music.

AKG wise is totally a different story. I always believe that the sound design of the headphones are related to the country of origin and their culture. AKG was from Austria where people are playing with instruments like Accordion which is more lively and light. Typical AKG headphones like K701, K702, are very refined, airy, sound stage is more backward, clean with thinner music body. Suitable for chamber music, piano, acoustic, new age, country music and so on. But I would say, if you were to choose AKG, has to be careful in choosing amp. In the market, more amps are using Sennheiser headphones as reference when they design it. If pairing with amp or dac that is not so suitable, it may sound bright and dry.

Hifiman wise, they are also good headphones that bring revolution.
HE500 and HE560 are really good and cheap, but you really need to invest in good amp.   
For your set up, I would say, can consider HE400i. It's consider clean in a dark background, refined, smooth and airy, thinner music body compare to sennheiser but much thicker than akg. The mid is fuller and overall with more bass compare to HE560, which makes it sound good with vocal. It's not a headphone that emphasise on bass, but it still has bass.
Overall performance, I rate it higher than Sennheiser HD6XX and AKG K7XX. Take Sennheiser bass as example, Senn is having the quantity but HE400i is having the quality, so refined and smooth with higher density.

I'm not sure how's the performance of your amp, as I haven't audition it before. But for all the headphones mentioned above, they need at least a decent amp to drive it, preferably desktop amp