What is the practical difference between the AdGuard app and just using the extension?
Practical differences: a) AdGuard for Mac is system wide - so it works with other browsers, email, newsfeeds, etc., b) can do a deeper inspection (and modification) of your web requests and responses, and c) do DNS blocking.
How it works difference: AdGuard for Safari is "just" a Safari extension using Apple's adblocker framework. AdGuard for Mac is a system extension that is able to inspect/modify all network traffic.
Management differences: AdGuard for Safari is easy - just enable and choose which filters. AdGuard for Mac is much more complex.
Trust: Because AdGuard for Mac can do more, you need to have greater trust that there is nothing bad going on. As a home user, I am happy with that. But, I would be careful to get sign off if I was using it within a large organisation.
AdGuard for Mac can intercept DNS requests and apply blocklists. Arguably you don't next NextDNS as well. I run AdGuard Home on a Pi as my DNS resolver, and it still blocks a few requests.