Both versions are the same except for one thing - the website version of PDF Expert for Mac doesn't have Apple Handoff. The app's updates are free and can be installed easily, both for Mac App Store and website versions. PDF Expert for Mac license allows you to install and use the app on up to three MacOS devices regardless of the Apple ID you're using. In the same time, Mac App Store version of PDF Expert for Mac can be installed on all macOS devices with the same Apple ID.
Instead of Apple Handoff functionality, we have Readdle Transfer, which is our own mechanism that allows to sync reading positions, annotations and edits made in PDF Expert across your Mac, iPad and iPhone.
You can use the following article for your reference: "Website or AppStore: Where To Buy PDF Expert?"