I am willing to develop an Outlook add-in for both, the web and the desktop version. My questions are:
If you are only updating the source code of the add-in (HTML, JavaScript, CSS) on the web server where you host it, then all users will get that right away since like Dmitry explained, Outlook will download the page like a browser would download it. So you should be careful updating the page of the production add-in - users will get the updates right away (although keep in mind that user's system/browser may cache previous version of the page for some time), so if you break the page or functionality of the add-in, users will get affected immediately, which may result in negative reviews for your add-in in the Office Store.
If you are updating the manifest XML file of the addin, then you need to publish new version of it to Office Store. Once it's published, that new version will automatically be updated for users who already installed an earlier version of the add-in. This automatic update will happen after they login to any of their Outlook clients that support add-ins.
Yes it is the same add-in for both web and desktop - that's the beauty of the new Office add-ins!
Hope this helps.