They're quite different. In particular, the Realm Mobile Platform is built on a full-blown database running on the user's device. One of Realm's biggest selling points is offline-first applications. As far as I can tell, this is not the case with Deepstream.
Just to build on what mwcampbell said (and in full disclosure, I'm on the team at Realm).
Architecturally, Realm and DeepstreamHub are quite different. Realm at its core is a distributed object database that updates changes to objects in realtime. DeepstreamHub, as far as I can tell, relies on a pub/sub and request/response architecture. They are two fundamentally different approaches to the challenge of realtime.
One of the biggest advantages with Realm is that you work with your objects in your native programming language, there's no JSON or ORMs to deal with, just your live objects that update automatically in the local datastore as changes are made on another sync'd client or on the server.
