Toolbar: The version of Plex Web included with the version of PMS installed on your system.
app.plex.tv: The latest version of Plex Web with latest features, fixes, etc.
When Plex releases a PMS update they don’t always include the latest version of Plex Web. Also, there are ongoing Plex Web releases, so the version of Plex Web on your server is usually not the latest release.
Example: I’m running PMS 1.14.0 which included Plex Web 3.69.1, so that is the version I get if I click on the Plex toolbar icon. The online version is 3.77.2, which is available via app.plex.tv.
You can check Authorized Devices to see which versions have been authorized by your PMS installation.
The Plex Web release announcement thread has info on the latest release.
The shortcut suggestion is just a way to put a bookmark on the desktop or taskbar. Making one for app.plex.tv is just a way to open Plex Web with less clicks than opening a browser, finding a bookmark, etc. You could just as easily make one for any web site/page, not just Plex.
