Downloading a local copy of One Note from Office 365 Online

You said you downloaded from onedrive but there's also an option on onenote online, so, for now, I'll treat it as if you had only tried onedrive but not the other (Microsoft is confusing, am I right?).

Onenote online provides an option to export onenote notebooks, but it is well hidden for some weird reason.

This answer is based on one of Microsoft's support articles: https://support.office.com/en-us/article/export-and-import-onenote-notebooks-a4b60da5-8f33-464e-b1ba-b95ce540f309?ui=en-US&rs=en-US&ad=US.

1. Go to https://www.onenote.com/notebooks and sign in with your Microsoft credentials.
2. You will see your notebooks. Right click the one you want to export.
3. There is an option Export notebook. Click that.
4. A popup window appears and you can click export to download locally (it also lets you choose where to save it).

Hope this helps, although I'm a little late.

Open OneNote, then open the OneNote file in the online location. Then click File > Export > Notebook and export the notebook to a local folder.

This copy will no longer be in sync with the online copy of the notebook.

Just to add that there is a distinction between OneDrive for personal use, and for corporate use. Business accounts do not allow to download notes in any way, none of the posts on this page were able to overcome this limitation.

From Microsoft help pages on OneNote app:

Exporting and importing notebooks through OneNote for the web is only available for notebooks stored on personal OneDrive accounts, not for notebooks stored on OneDrive for Business or SharePoint. For information about exporting notebooks to PDF files from OneNote 2016 for Windows, see Export notes from OneNote as a PDF.