Some great answers from others that cover a lot of ground. Here's a little bit extra.
The only advantage of WebSockets over plugins like Java Applets, Flash or Silverlight is that WebSockets are natively built into browsers and does not rely on plugins.
If by this you mean that you can use Java Applets, Flash, or Silverlight to establish a socket connection, then yes, that is possible. However you don't see that deployed in the real world too often because of the restrictions.
For example, intermediaries can and do shutdown that traffic. The WebSocket standard was designed to be compatible with existing HTTP infrastructure and so is far less prone to being interfered with by intermediaries like firewalls and proxies.
Moreover, WebSocket can use port 80 and 443 without requiring dedicated ports, again thanks to the protocol design to be as compatible as possible with existing HTTP infrastructure.
Those socket alternatives (Java, Flash, and Silverlight) are difficult to use securely in a cross-origin architecture. Thus people often attempting to use them cross-origin will tolerate the insecurities rather than go to the effort of doing it securely.
They can also require additional "non-standard" ports to be opened (something administrators are loathe to do) or policy files that need to be managed.
In short, using Java, Flash, or Silverlight for socket connectivity is problematic enough that you don't see it deployed in serious architectures too often. Flash and Java have had this capability for probably at least 10 years, and yet it's not prevalent.
The WebSocket standard was able to start with a fresh approach, bearing those restrictions in mind, and hopefully having learned some lessons from them.
Some WebSocket implementations use Flash (or possibly Silverlight and/or Java) as their fallback when WebSocket connectivity cannot be established (such as when running in an old browser or when an intermediary interferes).
While some kind of fallback strategy for those situations is smart, even necessary, most of those that use Flash et al will suffer from the drawbacks described above. It doesn't have to be that way -- there are workarounds to achieve secure cross-origin capable connections using Flash, Silverlight, etc -- but most implementations won't do that because it's not easy.
For example, if you rely on WebSocket for a cross-origin connection, that will work fine. But if you then run in an old browser or a firewall/proxy interfered and rely on Flash, say, as your fallback, you will find it difficult to do that same cross-origin connection. Unless you don't care about security, of course.
That means it's difficult have a single unified architecture that works for native and non-native connections, unless you're prepared to put in quite a bit of work or go with a framework that has done it well. In an ideal architecture, you wouldn't notice if the connections were native or not; your security settings would work in both cases; your clustering settings would still work; your capacity planning would still hold; and so on.
The only advantage of WebSockets over http streaming is that you don't have to make an effort to "understand" and parse the data received.
It's not as simple as opening up an HTTP stream and sitting back as your data flows for minutes, hours, or longer. Different clients behave differently and you have to manage that. For example some clients will buffer up the data and not release it to the application until some threshold is met. Even worse, some won't pass the data to the application until the connection is closed.
So if you're sending multiple messages down to the client, it's possible that the client application won't receive the data until 50 messages worth of data has been received, for example. That's not too real-time.
While HTTP streaming can be a viable alternative when WebSocket is not available, it is not a panacea. It needs a good understanding to work in a robust way out in the badlands of the Web in real-world conditions.
Are there any other significant differences that I am missing?
There is one other thing that noone has mentioned yet, so I'll bring it up.
The WebSocket protocol was designed to a be a transport layer for higher-level protocols. While you can send JSON messages or what-not directly over a WebSocket connection, it can also carry standard or custom protocols.
For example, you could do AMQP or XMPP over WebSocket, as people have already done. So a client could receive messages from an AMQP broker as if it were connected directly to the broker itself (and in some cases it is).
Or if you have an existing server with some custom protocol, you can transport that over WebSocket, thus extending that back-end server to the Web. Often an existing application that has been locked in the enterprise can broaden it's reach using WebSocket, without having to change any of the back-end infrastructure.
(Naturally, you'd want to be able to do all that securely so check with the vendor or WebSocket provider.)
Some people have referred to WebSocket as TCP for the Web. Because just like TCP transports higher-level protocols, so does WebSocket, but in a way that's compatible with Web infrastructure.
So while sending JSON (or whatever) messages directly over WebSocket is always possible, one should also consider existing protocols. Because for many things you want to do, there's probably a protocol that's already been thought of to do it.
I'm sorry if I am re-asking or combining many of the questions already on SO into a single question, but I just want to make perfect sense out of all the info that is out there on SO and the web regarding these concepts.
This was a great question, and the answers have all been very informative!
