Both of these cans are hungry for power and will need an amp. The 600 is a very neutral can and is the faster of the two. The 650 will give you a bit more bass. They are both classic and good cans (although not necessarily the best buys at current US MAP prices) if you want something in the neutral vein. If you want something bass heavy, look elsewhere.
Do you know if a fiio e7 would be enough for the HD600? Something with more juice would be necessary?
I'm a bit interested in the HD600 myself, I would almost certanly be using them at home with my Xonar STX, although I don't have a dedicated desktop amp I belive that the xonar can handle the HD600 for now.
Still, I would like to know if I can carry them for my grampa's house for example, but I only have the fiio e7 as portable amp.