A swap partition is a part of your storage disk that is formatted for swap use. It could also be it's own disk for high performance systems, but mostly for HPC.
A swap file is basically an empty disk image file that you mount as swap, the OS will use it just like a swap partition.
I prefer swap files because I find them easier to manage. I can easily delete, move, or enlarge the swap file whereas the partition will take a bit more work and is a bit riskier to change. Changing partition layouts can get very messy.
I always recommend a swap file be created when setting up a new Linux machine, even if you have loads of RAM. Some applications will use swap space to help performance, but I also like the fact that if I do something really dumb and fill up the root partition I can delete my swap file to free up space immediately, fix the full disk problem, and then recreate the swap file.