If you have a Firebase app open and you lose network connectivity, the app still continues to work fine and any modifications you make will be synced back to Firebase when you regain connectivity.

What we don't (quite yet) do is handle the case where you make changes, kill the app (without regaining network connectivity), restart the app, and then get network connectivity, but that's coming soon.

As for other differences, Dropbox Datastore seems to be tightly tied to Dropbox, so the end user for your app must have a Dropbox account (and if they want to collaborate with other users, those users must also have Dropbox accounts, etc.). So it's really for building apps on top of Dropbox.

Firebase and Parse are just generic backends, with no ties to other services, so your end users don't need anything to use your app.

As for differentiating Parse and Firebase, Firebase deals with data in real-time, pushing updates to apps as soon as data changes. Parse is a traditional request/response model, where your app has to explicitly "refresh" to get new data.