The free version of OneNote is lighter. The version that comes with Office is full-featured. (Don't know exactly what's different, I've only used the office version.)
The full version is also free. It comes included with the Office365 installer, but can be downloaded on its own:
https://www.onenote.com/download
OneNote syncs to OneDrive - so if you have O365 your storage space is quite large, whereas free users only have 5GB (though compared to Evernote's 25MB this is still plenty)
Unless they've changed it quite recently, the version that comes with Office365 and the one from https://www.onenote.com/download are different programs.
I did an office 365 install a week ago and recall a second Onenote popping up. Is it just no longer being developed, or did I get one of the last installs? I was unaware of the change!