What is the difference between software emulation (80GB) vs. backwards compatibility (60 GB)?

Answer 1

emulation is doing something in code that is normally done in hardware.

Backward compatible usually means the application can use the previous versions files.

Answer 2

it concerns the PS2 games you might want to play on it.
the backwards compatible edition has a dedicated PS2 chip in it so all the PS2 games run fine.
the software emulation edition took it out (probably to reduce costs) and uses the PS3 to pretend it's a PS2. this is more difficult and as such not all PS2 games will work on it at present (though Sony are releasing downloads periodically which improve things as they sort them out)
if you still have a PS2 or are just interested in the PS3 part then get the software emulation version (the bigger hard drive may come in handy down the line)

here in the UK though we have a choice of software emulation (60GB) and .........