Why put things other than /home to a separate partition?

1. Minimizing loss: If /usr is on a separate partition, a damaged /usr does not mean that you cannot recover /etc.
2. Security: / cannot be always ro (/root may need to be rw etc.) but /usr can. It can be used to make ro as much as possible.
3. Using different FS: I may want to use a different system for /tmp (not reliable but fast for many files) and /home (has to be reliable). Similary /var contains data while /usr does not so /usr stability can be sacrifice but not so much as /tmp.
4. Duration of fsck: Smaller partitions mean that checking one is faster.
5. Mentioned filling up of partions, although other method is quotas.

A separate /usr can be useful if you have several machines sharing the same OS. They can share a single central /usr instead of duplicating it on every system. /usr can be mounted read-only.

/var and /tmp can be filled up by user programs or daemons. Therefore it can be safe to have these in separate partitions that would prevent /, the root partition, to be 100% full, and would hit your system badly. To avoid having two distinct partitions for these, it is not uncommon to see /tmp being a symlink to /var/tmp.

Because ordinary users can cause things to be written to /var and /tmp, and thus potentially cause problems for the whole system. This way user processes can fill up /var and /tmp, but not the root fs. A separate /usr is useful for /usr over NFS, or other remote fs.

(I hope this is clear, I haven't had any coffee yet)