Unseen, your confusion is warranted as the site does a poor job of outlining the practical differences between the two. You're right in that it appears both do just about the same thing. There are a few subtle differences that I've been able to identify however. (Note I only have experience using Teamviewer on a Mac to control another Mac but my guess is it shouldn't make much of a difference overall)
Teamviewer host has a bit deeper integration with your system and runs in the background similar to how Dropbox automatically runs in the background. It doesn't have the usual application window that opens on screen or an app icon active in the dock when it's running (all you see is a small badge nested up by your clock). This keeps its presence more discreet. This should also help keep a user of your target computer from accidentally quiting the service, and in turn cutting off your access, if they see it running on their dock like they do with the standard teamviewer product.
If you want to be able to discreetly control a computer without anything getting in the way or confusing anyone that uses the target computer, I'd opt for installing Teamviewer Host. In your personal scenario this is what I'd recommend. For another example use case, I've installed host on my parents' computer so I can easily help them out with any computer problems and periodically check in on their system. They'll never really see much or need to do anything on their end. Once set up, it'll just work in the background. No hassle or confusion for them.
In contrast the standard Teamviewer product will open an application window on your screen and has an app icon in the dock when it's running. Install this version if you don't mind the application feeling more present on the computer or if you plan to utilize the meeting and other functionalities it offers.
(While I don't have experience with it, I do also believe you have a deeper level control with Teamviewer Host and can even access a device once it's powered on even before a user has logged in. That's a whole other more technical rabbit hole I'll leave for someone else.)