Websockets have a couple of big downsides in terms of scalability that ajax avoids. Since ajax sends a request/response and closes the connection (..or shortly after) if someone stays on the web page it doesn't use server resources when idling. Websockets are meant to stream data back to the browser, and they tie up server resources to do so. Servers have a limit in how many simultaneous connections they can keep open at one time. Not to mention depending on your server side technology, they may tie up a thread to handle the socket. So websockets have more resource intensive requirements for both sides per connection. You could easily exhaust all of your threads servicing clients and then no new clients could come in if lots of users are just sitting on the page. This is where nodejs, vertx, netty can really help out, but even those have upper limits as well.
Also there is the issue of state of the underlying socket, and writing the code on both sides that carry on the stateful conversation which isn't something you have to do with ajax style because it's stateless. Websockets require you create a low level protocol which is solved for you with ajax. Things like heart beating, closing idle connections, reconnection on errors, etc are vitally important now. These are things you didn't have to solve when using AJAX because it was stateless. State is very important to the stability of your app and more importantly the health of your server. It's not trivial. Pre-HTTP we built a lot of stateful TCP protocols (FTP, telnet, SSH), and then HTTP happened. And no one did that stuff much anymore because even with its limitations HTTP was surprisingly easier and more robust. Websockets bring back the good and the bad of stateful protocols. You'll learn soon enough if you didn't get a dose of that last go around.
If you need streaming of realtime data this extra overhead is warranted because polling the server to get streamed data is worse, but if all you are doing is user interaction->request->response->update UI, then ajax is easier and will use less resources because once the response is sent the conversation is over and no additional server resources are used. So I think it's a tradeoff and the architect has to decide which tool fits their problem. AJAX has its place, and websockets have their place.
Update
So the architecture of your server is what matters when we are talking about threads. If you are using a traditionally multi-threaded server (or processes) where a each socket connection gets its own thread to respond to requests then websockets matter a lot to you. So for each connection we have a socket, and eventually the OS will fall over if you have too many of these, and the same goes for threads (more so for processes). Threads are heavier than sockets (in terms of resources) so we try and conserve how many threads we have running simultaneously. That means creating a thread pool which is just a fixed number of threads that is shared among all sockets. But once a socket is opened the thread is used for the entire conversation. The length of those conversations govern how quickly you can repurpose those threads for new sockets coming in. The length of your conversation governs how much you can scale. However if you are streaming this model doesn't work well for scaling. You have to break the thread/socket design.
HTTP's request/response model makes it very efficient in turning over threads for new sockets. If you are just going to use request/response use HTTP its already built and much easier than reimplementing something like that in websockets.
Since websockets don't have to be request/response as HTTP and can stream data if your server has a fixed number of threads in its thread pool and you have the same number of websockets tying up all of your threads with active conversations, you can't service new clients coming in! You've reached your maximum capacity. That's where protocol design is important too with websockets and threads. Your protocol might allow you to loosen the thread per socket per conversation model that way people just sitting there don't use a thread on your server.
That's where asynchronous single thread servers come in. In Java we often call this NIO for non-blocking IO. That means it's a different API for sockets where sending and receiving data doesn't block the thread performing the call.
So traditional in blocking sockets when you call socket.read() or socket.write() they wait until the data is received or sent before returning control to your program. That means your program is stuck waiting for the socket data to come in or go out until you can do anything else. That's why we have threads so we can do work concurrently (at the same time). Send this data to client X while I wait on data from client Y. Concurrencies is the name of the game when we talk about servers.
In a NIO server we use a single thread to handle all clients and register callbacks to be notified when data arrives. For example
socket.read( function( data ) {
// data is here! Now you can process it very quickly without waiting!
});
The socket.read() call will return immediately without reading any data, but our function we provided will be called when it comes in. This design radically changes how you build and architect your code because if you get hung up waiting on something you can't receive any new clients. You have a single thread you can't really do two things at once! You have to keep that one thread moving.
NIO, Asynchronous IO, Event based program as this is all known as, is a much more complicated system design, and I wouldn't suggest you try and write this if you are starting out. Even very Senior programmers find it very hard to build a robust systems. Since you are asynchronous you can't call APIs that block. Like reading data from the DB or sending messages to other servers have to be performed asynchronously. Even reading/writing from the file system can slow your single thread down lowering your scalability. Once you go asynchronous it's all asynchronous all the time if you want to keep the single thread moving. That's where it gets challenging because eventually you'll run into an API, like DBs, that is not asynchronous and you have to adopt more threads at some level. So a hybrid approaches are common even in the asynchronous world.
The good news is there are other solutions that use this lower level API already built that you can use. NodeJS, Vertx, Netty, Apache Mina, Play Framework, Twisted Python, Stackless Python, etc. There might be some obscure library for C++, but honestly I wouldn't bother. Server technology doesn't require the very fastest languages because it's IO bound more than CPU bound. If you are a die hard performance nut use Java. It has a huge community of code to pull from and it's speed is very close (and sometimes better) than C++. If you just hate it go with Node or Python.