What is the use for the "realtime" priority for a process?

Exactly what you describe, at the cost of slowing the rest of your computer down to a crawl it runs the process in real-time giving it a higher priority than everything else. This is only necessary for time-critical applications, most people don't see these in their daily life...

This may seem unnecessary but when a process is started, parameters can be passed that set the process to run at various levels of priority. The Task Manager shows this as an optional setting because it is an option that can be set for the process by the software that initiated it's execution. Most software runs under a "Normal" level, but occasionally you may see a few that are running at a lower level by default. I've never seen software that starts in an above normal state.

It is also worth mentioning that there is a case when a process can run at the highest priority setting and it not seem to greatly affect your system performance. For instance, if a process is started and it is assigned to work on only 1 of, say, 8 CPU cores, setting the processes priority to "real time" would mean that it would consume one processor, leaving 7 to be used by the OS and any other software. In this case, real time execution could have almost no affect on the system but give a reasonable performance bump to the high priority process.

I would almost never recommend manually setting a process to run in "real time" from within the System Manager. This is because you are changing the execution settings for a process that may have been designed to operate under one specific execution state. Changing the execution priority can, in some cases, crash the application and if it's a data-write operation, possibly corrupt your data!

However, there are some cases where using a maximum execution priority setting for a process can be the right thing to do, but even then, such priority settings should be set by the controlling application and not by the end user through a system management dialog.

I, personally, would recommend NEVER changing a tasks priority unless you really don't care if it crashes and the task is COMPLETELY non-critical. If software documentation explicitly states that you can elevate the process to run at a faster speed in order to expedite a very large load of processor work, then that would be acceptable as well, however, I'd be a bit skeptical if a mechanism from within the application is not provided.

As a personal anecdote, there is only one process that I occasionally change manually-- software compression applications. I haven't had to unzip a massive file in a long time, but I have found that if I am in rush (ahem being impatient) I can crank up the process priority and get a moderate bump in performance. Since these applications start a work task, and then close when they are finished, I don't have to worry about unsetting my processor settings because when the application is finished extracting the data, it will stop on it's own. This is the only time that I have found it personally useful to control the processor speed through system management.