What is the advantage of being able to select the MONO button?

Ideally, one would listen to records mixed and mastered as mono recordings on a mono setup and stereo recordings on a stereo set up.  Listening to mono recordings through two speakers of a stereo setup can sound very good too.  When  listening to stereo recordings in a mono setup with just one speaker, the two channels have to be summed to one.  This is not ideal because a recording engineer would never simply sum stereo channels to do a mono mix.  Still, that is what has to be done so it is best to do it correctly.  A local shop which builds and sells some mono horn speaker systems also builds and sells a box with one or two transformers for doing the summing.  This sounds MUCH better than a Y connector or.  Their tech made a box that held a selection of different transformers and a Y connection so one could do a quick A-B comparison.  There were easily discernible differences, and to me, there were good modern and vintage transformer alternatives (the Y connection was way behind in my opinion).

There are also issues with how sounds cancel coming from 2 speakers vs. 1 due to head related effects.  The ideal playback for mono recordings really is just 1 speaker, or two speakers right next to each other.

I used to play chess with my former next-door-neighbor. I'd set up a "chess table" sideways in front of my stereo. I'd set the system to mono so we could each hear all of the music instead of just what was coming out of the speaker we were next to.

Thanks, but none of that addressed my concerns. 

Anyone?

I have assembled a dedicated mono rig (VPI Super Scoutmaster Special Edition w JMW Memorial tonearm, Ortofon Quintet mono cart into Plinius P-100 phono-pre with no mono button). I believe this to be described as a true mono cart and I’m getting different ideas as to what that means. Does this cart require a pre with mono button or is that just for playing mono records on a stereo rig?

According to the manufacturer:

Quintet Mono uses a strapped output to deliver the same output signal from both sets of pole pins. This effectively eliminates the need for mono-specific equipment, making it possible to enjoy true mono reproduction on any stereo playback system.

You don’t need a preamp with a "mono" button, it will play in mono through any stereo equipment and both speakers will output the same content.

What exactly does this button do if I had a preamp that had one?

The function of the "Mono" button in the preamp is to sum the left and right stereo signals into a mono signal for the speaker, so both speakers will output the same content.

What is all this about Y adapters? Do I need anything like this to optimize this set up?

No, I do not recommend using a Y adapter for the preamp output, it shorts the two outputs together and may damage the preamp.