It is the same Dock
The two Mac downloads differ only in who bills you and where updates come from. The web app is the one that works differently: it runs on our servers, so your store credentials are kept there instead of on your Mac, and not every feature is available.
One licence, the Mac and the browser
Buy the download and the web app is included, and the other way round: it is one subscription. The App Store stays separate: a subscription bought there unlocks neither of the two, and a licence key does not unlock the App Store version.
Three Macs on one licence
The direct version activates on up to three Macs, and you can free one up whenever you like. On the App Store, Apple's own rules apply.