Correct, there is no difference in the code between the hosted version and the open source version. However, the hosted version does come with support ticket access and scheduled upgrades are free as part of the monthly fee.
We also handle set-up/install, as well as scheduled backups of the data both locally and externally.
The recommended method of sharing sessions between two separate and distinct applications is through OAuth2. We recommend this approach because NodeBB maintains its own user records, so that we can keep track of user-related metrics and other data. Relying on another database would be tricky, prone to breaking, and quite possibly dangerous.
Luckily, it's quite straightforward to get things working with OAuth2!
The first step is getting your application to expose an OAuth2 endpoint. If you're running a Node.js based app, you can use a module called OAuth2orize.