What is the difference between 'git format-patch and 'git diff'?

From the manuals git-format-patch prepares patches suitable for email submission, while git-diff shows changes.

They are two different things and have different purposes, they just happen to output a patch format. But git-format-patch adds data about a commit (date, author, commit message) and bundles it up into a format that is suitable for sending as a Unix mail message (although these are just files, so they can be sent to other methods and still applied by git-am).

Also git-format-patch generates a patch file for each commit in the range that you specify. These changes will be added as commits to your repository with git-am.

git-diff just shows the diff between the two states you ask for, and can be used to create a patch file. But this is just a normal patch file and applying the patch will just change the state of the working directory.

And yes, you can create a patch for your index that way.

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.