A patch created with git format-patch will also include some meta-information about the commit (committer, date, commit message, …) and will contains diff of binary data. Everything will be formatted as a mail, so that it can be easily sent. The person that receive it can then recreate the corresponding commit with git am and all meta-data will be intact. It can also be applied with git apply as it is a super-set of a simple diff.
A patch crated with git diff will be a simple diff with context (think diff -u). It can also be applied with git apply but the meta-data will not be recreated (as they are not present).
In summary, git format-patch is useful to transmit a commit, while git diff is useful to get a diff between two trees.