There actually a couple of interesting dimensions to consider when comparing GTFS-realtime to the various RESTful web-interfaces that agencies have cooked up vs other real-time data standards (eg. SIRI):
Bulk vs piece-wise access: GTFS-realtime is designed for bulk access to real-time data (eg. what's the status of every vehicle in the system right now?). By comparison, many RESTful APIs provided by agencies are geared towards piece-wise access (eg. what's the status of the next couple of buses arriving at a particular stop?). The RESTful API design reflects the typical applications consuming the API: station arrival boards, mobile apps giving next-bus-at-my-stop status, etc. Those applications are great and provide immediate, obvious benefits to riders, so it's no surprise why agencies provide these kinds of APIs. However, piece-wise APIs make some kinds of applications tricky to write. For example, building a real-time trip planner using a piece-wise API is really tricky, since you really need a full picture of the system to do real-time routing. Bulk access specifications like GTFS-realtime or SIRI (ET, VM, SX) more easily and efficiently support these types of applications. Generally speaking, anything you can do with a high-level piece-wise API can also be done with a bulk-access API, but it might require a bit more effort on the part of the programmer.
Markup language: Many real-time data formats use an existing markup language to encode data - XML, JSON, Protocol Buffers, or perhaps one of their own invention. Most RESTful APIs tend to be XML or JSON. Protocol Buffers are used by GTFS-realtime. Each has different trade-offs in terms of ease-of-use, verbosity, encode-decode speed, size, etc. These trade-offs tend to become more evident when framed by the "bulk vs piece-wise" access tradeoff.
Semantics: Though you can spend life-times arguing the merits of XML vs JSON vs ProtoBufs vs other encoding schemes, when you get down to it, most of the actual semantic data coming from real-time systems looks largely the same. At end of the day, they're all talking about the same underlying systems, so maybe that's not surprising. As result, GTFS-realtime Trip Updates look a lot like SIRI Vehicle Monitoring message. Ditto GTFS-realtime Alerts and SIRI Situation Exchange messages. Many RESTful APIs look a lot like SIRI Stop Monitoring messages. And all the static discover APIs look a lot like the data you get from a static GTFS feed (or maybe a NeTEx message at some point?).
tl;dr - There are different trade-off in real-time API design. Almost every agency / real-time vendor in North America who publishes an API has approached those trade-offs differently and they've all got slightly different APIs as result (hurray!). GTFS-realtime and some of the RESTful SIRI work that Mike Frumin has championed at NYC MTA are efforts to bring order to the madness, but they are new kids on the block so time will tell.
Me personally? I want GTFS-realtime because it gives me the most flexibility to write cool transit apps and run them everywhere. RESTful next-bus-at-stop APIs have their place, but the journey doesn't stop when your bus arrives to pick you up. Having access to information about the entire transit system is critical for giving riders up-to-date information about their entire trip, not just the start.