ajax / html request
Slowpoke. Good for loading pages without navigating to another url.
No good for action games but should be fine for turn based games I imagine. That is unless it's a very fast paced turn based game (think roguelike fast).
The good thing is that the server only needs to handle one request at a time.
So the server code is serial and not concurrent. This means it's a lot less complicated.
Each request is often processed in an infinitely short amount of time. I think the default limit is something like 15 seconds. The point being the server grabs a request, starts working on it and only it, and is done after a short while.
Fast and responsive, great for action games.
The server needs to handle all clients simultaneously (concurrently) and ceaselessly. This means possibly more processing power and more memory. This means that you are running a process that doesn't stop supposedly ever (PHP isn't oft for such processes). You need a reasonably powered machine with serious bandwidth capability. This is considered more resource intensive.
The code is also far more complex and often requires threads or chunking up the work into small portions and supporting multiple user request in parallel. Both threads and chunking up the workload into small pieces make the code somewhat more complicated and harder to debug.
Please be aware that this means the devs need prior experience with either threads or what is basically an implementation of threads as a serial program. Also considering that most server CPU's nowadays have a bunch of cores, using threads is possibly preferred.
Generally, an indie socket based game + server could cost could somewhere in the tens of thousands of dollars to develop. This is because of the graphics and client-side programming any game would require plus the added costs of developing a stable server.