Submodule can be used when you want to nest a private repository inside of a public one. Only authorized users will be able to fetch from the submodule repository. This can be useful in some cases, such as storing private keys in an otherwise public project.
Also, if you have a very large submodule repository, it won't inflate the size of the repository that contains it.
git-subtree can flatten the other tree into a single commit while preserving the ability to use all its other functionality, which is actually probably a space-saving measure compared to submodules when you consider that anyone who uses your repo probably needs to get the submodule anyway, and that means downloading that other giant repo in its entirety vs. downloading a flattened copy of it.
So if I understand this correctly, it is more of a convenient way to copy (and update) another repository into another one, instead of being a reference to another repository.
To respond to the comment about this being useful for private keys: if this is the case, does this also mean unauthorized users will not be able to checkout the submodule at all? Or will they just get a copy of the files (which would not make sense with private keys).
When you git clone a repository with a submodule, before you call git submodule sync/update and all that jazz, you just have an empty directory where the submodule should be.
Also, if you have a very large submodule repository, it won't inflate the size of the repository that contains it.