The site in the PWA is the same as the browser version, but it behaves like an app in the task switcher and has an app icon. The previous apps available in the app store / Google Play store were basically just "wrappers" of the browser version of the site anyway, and those had some downsides...
I don't think they are, because I'm pretty sure that one must first grant permission for them to be sent after pressing "Enable push notifications". Perhaps permission was previously granted in Chrome, though I'm not sure.
Though I don't think the old app has anything which the PWA doesn't, and the PWA supports push notifications, while I can't seem to get them to work when accessing the site via the old app. The PWA also seems a lot faster.
I don't think it gets in the way of the other buttons in the off-canvas menu. It also goes away if the PWA is installed, even if a regular browser is still being used to view the site rather than the PWA.
If you have an Android phone you can now install what's called a Progressive Web App, which provides the functionality of an app without the issues that the previous apps had.
In Chrome on Android, you should see a button to install it in the off-canvas menu:
Press it, confirm that you want to...