As mentioned, if you want a “per-branch stash,” you really want a new branch forking off from the existing branch.
Also, besides the already mentioned fact that the stash allows you to pull into a branch that you’re working on, it also allows you to switch branches before you have committed everything. This is useful not for cherry-picking in the usual sense so much as for cherry-picking your working copy.
F.ex., while working on a feature branch, I will often notice minor bugs or cosmetic impurities in the code that aren’t relevant to that branch. Well, I just fix those right away. When time comes to commit, I selectively commit the relevant changes but not the fixes and cosmetics. Instead I stash those, which allows me to switch to my minor-fixes-on-stable branch, where I can then apply the stash and commit each minor fix separately. (Depending on the changes in question, I will also stash some of them yet again, to switch to a different feature branch, where I apply those.)
This allows me to go deep into programming mode when I am working, and not worry about proper librarianship of my code. Then when I take a mental break, I can go back and carefully sort my changes into all the right shelves.
If the stash weren’t global, this type of workflow would be far more difficult to do.
git-stash is most useful to me to move not-yet-checked-in changes off to a different branch than the one that is currently checked out.
For example - I often find myself doing simple changes on a bug-fixes branch; only to find that a change I'm working on is more complex than I first guessed. Git-stash is the easiest way to move that set of changes to a different branch.