Linux single boots with the prompt having full root powers but with no networking and the system mounted normally, I think. It has been a couple of years since I have used linux single.
Rescue boots into a kernel and bash, nothing mounted (some distros differ) and everything is done manually, mounting drives, mounting the /proc directory to a mounted system so that you can chroot into it and actually do processes such as compiling. I tend to use rescue and do stuff by hand, myself.
I haven't used emergency mode, so I am unfamiliar with it.
It just depends on what you need to do and learning the fastest and easiest ways to do what needs to be done.