I've been an avid Chromebook user since they were released. I carry with me an Asus Chromebook every day and use it more frequently than even my full Windows laptop. As a OneNote user it was important to find out how well I could take advantage of OneNote on the Chromebook and what features I'd lose in the limited online version.
OneNote Online is the tool
When using OneNote on a Chromebook you're limited to using the OneNote Online version of the application. This means right away many of the features you may be used to on the desktop version (such as custom tags) are not available. However, if you use OneNote with these limitations in mind you will find it to be a useful note taking and personal information management tool.
Screen clipping requires a plug-in
I use the screen clipping tool on OneNote often, more often than any other on my computers. However, since you do not have the desktop capability for screen clipping with OneNote Online, you need to take other steps. One of the best things is to install the extension Clip to OneNote in your Chrome browser. This extension allows for capturing entire pages, sections, articles, and even products directly to OneNote notebooks.
One Notebook at a time
Unlike the desktop, when you're using the Online version you can only have one notebook open at a given time. However, you can get around this limitation by two-finger clicking (press the left and right mouse buttons at the same time) on the name of a notebook and then tell Chrome to open the notebook in a new tab. You can do the same thing by holding down the CTRL key while left-clicking on a notebook name.
Search is limited
If you use the search box on the Notebooks page it is limited to the names of the notebooks, not the content of the notebooks as you find on the desktop version. If you use the search box in OneNote it will default to searching the page you are currently viewing. However, if you press Ctrl-E you can tell OneNote to search the entire section you are in. When OneNote Online is your primary tool, you'll want to set up your notebooks and sections in such a way so you can leverage search in the most effective manners possible.
Another workaround for searching across notebooks is to click on Manage and Delete at the notebook page and then use the search in OneDrive to search all the contents of your notebooks. This needs to be integrated into the OneNote Online tool by Microsoft, but for now it at least gives you a chance to find your content.
It's not perfect but it's not terrible
For now, OneNote Online is a feature limited version of it's big brother. Whether we'll see more of the desktop capabilities migrate to the online version only time will tell. All in all OneNote Online is an excellent addition to the collection of tools you can use to take advantage of the power and flexibility of your Chromebook.
UPDATE - September 2017
Microsoft has been busy adding features and improvements to the online version of OneNote. Recently they have been upgrading the look and feel of the interface, making it much more visually appealing for daily use. While I'm still waiting on some core features to make it over from the desktop (custom tags, templates, etc.) you can definitely see OneNote is getting a lot of attention in Redmond.
Update - October 2017
With the release of the Google Pixelbook and it's touchscreen / pen combination, I'm expecting big things when it comes to OneNote functionality on Chromebooks. There's no doubt it's becoming a stronger contender in the education space with recent deployments and upgrades, but the addition of higher end Chromebooks to the ecosystem can only help drive adoption that much more.
Update - November 2017
Microsoft has just released the Microsoft Office suite for use on Chromebooks through the Google Play store. This includes Word, Excel, and PowerPoint (the Android versions). Alas, this does NOT include OneNote. C'mon guys...get on the stick.
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