Rope, but clean it regularly and try to keep it out of the UV light if possible. I made a quick cheap and easy rope cleaner a little while ago.= out of the guts from a broken hose nozzle and a short piece of 32mm conduit. (an no, thats not my winch rope, just a piece of rope to demonstrate)
As for type/price, not sure I will weigh into that debate, except to say make sure replacement parts are available and as mentioned above if its to pull you out of trouble on a long solo trip then prob spend a little more!
Personally I bought one of the old Tigerz 10,000lb winches with the external brake (not in the drum like most so perfect for rope as no heat), and very happy with my purchase.