EIS of the 767-200 was with the JT9D-74RD/7R4E engine in 1982, quickly followed by the CF6-80A/-80A2. The longer 767-300 version followed a few years later, again with both engine types. Max thrust for both engine types was about 50,000 lbs.
The PW4000 and CF6-80C2 were certified in the 1988 time frame. The -80C2 was initially the non-FADEC 'PMC' version, followed about a year later by the FADEC, with the RB211-524G/H about a year after that. All three engine types had the advantage of more thrust (~62k for the PW, ~60k for the CF6-80C2B6 and RB211-524H) and significantly better fuel burn than the JT9D and CF6-80A engines. Most of the JT9D and CF6-80A powered 767s were converted to freighters years ago (and many retired by now).
As I noted earlier, the PW4000, CF6-80C2, and RB211-524G/H engines are common between the 767 and the 747-400. Interesting tidbit, initially GE resisted the FADEC version of the CF6-80C2 - they didn't feel the advantages of FADEC outweighed the costs (and the PMC version was quite a nice hydro control). Boeing said 'fine, but were not putting throttle cables in the 747-400 - if you want to be on the -400 you need to go FADEC'.
A380 - when I said there is currently little to choose between the CF6-80C2 and the PW4000, it's across the board - maintenance, fuel burn, etc. It's really down to what the operator wants with neither engine having a clear advantage.