Why use AJAX when WebSockets is available?
Most of the complaining I have read about websockets and security is from security vendors of web browser security and firewall security tools. The problem is they don't know how to do security analysis of websockets traffic, because once it has done the upgrade from HTTP to the websocket binary protocol, the packet content and its meaning is application specific (based on whatever you program). This is obviously a logistic nightmare for these companies whose livelihood is based on analyzing and classifying all your internet traffic. :)
WebSockets isn't intended to replace AJAX and is not strictly even a replacement for Comet/long-poll (although there are many cases where this makes sense).
The purpose of WebSockets is to provide a low-latency, bi-directional, full-duplex and long-running connection between a browser and server. WebSockets opens up new application domains to browser applications that were not really possible using HTTP and AJAX (interactive games, dynamic media streams, bridging to existing network protocols, etc).
However, there is certainly an overlap in purpose between WebSockets and AJAX/Comet. For example, when the browser wants to be notified of server events (i.e. push) then Comet techniques and WebSockets are certainly both viable options. If your application needs low-latency push events then this would be a factor in favor of WebSockets. On the other hand, if you need to co-exist with existing frameworks and deployed technologies (OAuth, RESTful APIs, proxies, load balancers) then this would be a factor in favor of Comet techniques (for now).
If you don't need the specific benefits that WebSockets provides, then it's probably a better idea to stick with existing techniques like AJAX and Comet because this allows you to re-use and integrate with a huge existing ecosystem of tools, technologies, security mechanisms, knowledge bases (i.e. far more people on stackoverflow know HTTP/Ajax/Comet than WebSockets), etc.
On the other hand, if you are creating a new application that just doesn't work well within the latency and connection constraints of HTTP/Ajax/Comet, then consider using WebSockets.
Also, some answers indicate that one of the downsides of WebSockets is limited/mixed server and browser support. Let me just diffuse that a bit. While iOS (iPhone, iPad) still supports the older protocol (Hixie) most WebSockets servers support both Hixie and the HyBi/IETF 6455 version. Most other platforms (if they don't already have built-in support) can get WebSockets support via web-socket-js (Flash based polyfill). This covers the vast majority of web users. Also, if you are using Node for the server backend, then consider using Socket.IO which includes web-socket-js as a fallback and if even that is not available (or disabled) then it will fall back to using whatever Comet technique is available for the given browser.
Update: iOS 6 now supports the current HyBi/IETF 6455 standard.
In addition to issues with older browsers (including IE9, as WebSockets will be supported starting from IE10), there are still big problems with network intermediaries not yet supporting WebSockets, including transparent proxies, reverse proxies, and load balancers. There are some mobile carriers that completely block the WebSocket traffic (that is, after the HTTP UPGRADE command).
With years passing, WebSockets will be more and more supported, but in the meantime you should always have an HTTP-based fall-back method for sending data to the browsers.
Fast forward to December 2017, Websockets are supported by (practically) every browser and their use is very common.
However, this does not mean that Websockets managed to replace AJAX, at least not completely, especially as HTTP/2 adaptation is on the rise.
The short answer is that AJAX is still great for most REST applications, even when using Websockets. But god is in the details, so...:
AJAX for polling?
The use of AJAX for polling (or long polling) is dying out (and it should be), but it still remains in use for two good reasons (mainly for smaller web apps):
For many developers, AJAX is easier to code, especially when it comes to coding and designing the backend.
With HTTP/2, the highest cost related to AJAX (the establishment of a new connection) was eliminated, allowing AJAX calls to be quite performant, especially for posting and uploading data.
However, Websocket push is far superior to AJAX (no need to re-authenticate or resend headers, no need for "no data" roundtrips, etc'). This was discussed a number of times.
AJAX for REST?
A better use for AJAX is REST API calls. This use simplifies the code base and prevents the Websocket connection from blocking (especially on medium sized data uploads).
There are a number of compelling reasons to prefer AJAX for REST API calls and data uploads:
The AJAX API was practically designed for REST API calls and it's a great fit.
REST calls and uploads using AJAX are significantly easier to code, both on the client and the backend.
As data payload increases, Websocket connections might get blocked unless message fragmentation / multiplexing logic is coded.
If an upload is performed in a single Websocket
sendcall, it could block a Websocket stream until the upload had finished. This will reduce performance, especially on slower clients.
A common design uses small bidi messages transferred over Websockets while REST and data uploads (client to server) leverage AJAX's ease of use to prevent the Websocket from blocking.
However, on larger projects, the flexibility offered by Websockets and the balance between code complexity and resource management will tip the balance in favor of Websockets.
For example, Websocket based uploads could offer the ability to resume large uploads after a connection is dropped and re-established (remember that 5GB movie you wanted to upload?).
By coding upload fragmentation logic, it's easy to resume an interrupted upload (the hard part was coding the thing).
What about HTTP/2 push?
I should probably add that the HTTP/2 push feature doesn't (and probably can't) replace Websockets.
This had been discussed here before, but suffice to mention that a single HTTP/2 connection serves the whole browser (all the tabs/windows), so data being pushed by HTTP/2 doesn't know which tab/window it belongs to, eliminating it's capacity to replace Websocket's ability to push data directly to a specific browser tab / window.
While Websockets are great for small bi-directional data communication, AJAX still carried a number of advantages - especially when considering larger payloads (uploads etc').
And Security?
Well, generally, the more trust and control is offered to a programmer, the more powerful the tool... and the more security concerns that creep up.
AJAX by nature would have the upper hand, since it's security is built in to the browser's code (which is sometimes questionable, but it's still there).
On the other hand, AJAX calls are more susceptible to "man in the middle" attacks, while Websockets security issues are usually bugs in the application code that introduced a security flaw (usually backend authentication logic is where you'll find these).
Personally I don't find this to be that big of a difference, if anything I think Websockets are slightly better off, especially when you know what you're doing.
My Humble Opinion
IMHO, I would use Websockets for everything but REST API calls. Big data uploads I would fragment and send over Websockets when possible.
Polling, IMHO, should be outlawed, the cost in network traffic is horrid and Websocket push is easy enough to manage even for new developers.