How do I find out what is using up all the space on my / partition?

It's entirely possible that you have a very large deleted file (or lots of little ones) that a process still has an open file handle on. The way to find them is to run

# lsof | grep "deleted"

If you see lots of lines that end with "(deleted)" then you can find the process Id that has them open and restart it. Once that happens, your disk space should return.

If this does not fix it, then I'd recommend a fsck.

There are a bunch of reasons du does not equal df. See the answers to this question.

Some are overlay mounts, lots of small files and a larger block size, and deleted files that are still in use. Overlay mounts are when you mounted a filesystem on a mount point that had files in it, so du doesn't see them.

The main difference between the two is that df just checks the superblock and trusts it, where as du scans all the files that it is able to see, and adds them up. See this IBM link for information on the superblock.

Always use the -x option with du when you are chasing problems like this. It keeps du from crossing filesystems.