Does it do anything more than what Rectangle covers? For instance, can it switch focus between screens or applications within a screen with a keyboard shortcut?

So hammerspoon expands way past windows management to all kinds of automation and interaction with the operating system. See a list of plugins here:

https://www.hammerspoon.org/Spoons/

The window management plugin I just linked you doesn’t really have any additional functionality over rectangles but the implementation is better. My BTT set up was similar to Rectangles in that there was basically a different keyboard shortcut for each position on the screen. With hammerspoon you just use your modifier key and the arrows. So say your app was full screen if you do hyper + left arrow it will take up half the screen on the left hand side. Press hyper + left arrow again and it will now take up 1/3. Press hyper + right arrow and it will expand back out to 1/2 again. Press hyper + left + right and it will fill the entire width of the screen. Same rules apply for up and down. It’s really intuitive and you don’t have to bother remembering what all the keyboard shortcuts were to get things where you want them.

As for the switch focus between screens I believe there is a shortcut for this but I haven’t used it yet. There are native macOS shortcuts though, cmd + ~ will navigate between different screens of the same application, cmd + tab for navigating between applications and cmd + shift + [ for navigating between tabs.

The cmd + ` is limited to switching with the same application. However, another command, "Move focus to active or next window", by default bound to `Ctrl + F4`, cycles through all applications on the active workspace, which I have started to use extensively. Took me some time to find it out.