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).

The mono button does what the Y connector does to sum left and right.  You can then use the amp to feed identical summed signal to the two speakers of a stereo system or use one channel to feed a single speaker.  But, as I opined above, summing with a Y connector or a mono button is inferior to using a transformer to sum to mono.  You can google about the specifics of using a transformer for doing mono.  
 

I don’t have/want a mono system but I’ve heard many that are quite good.  Some mono systems were installed for practical reasons—e.g., a kitchen system where a listener can be in many different locations and still hear all of the music.  

You don’t “need” a mono button to play mono records.  Whether or not playing a mono record using information from the sum of two identical channels vs. just one channel is a different matter (I have no idea).  You need to sum two stereo channels to play stereo records in a mono set up.  Not everyone who has a mono setup only listens to mono recordings in such systems.

A true mono cartridge has on one signal generating element that responds only to side to side movement of the stylus and ignores vertical movement (which is information in stereo but just noise in mono).  Modern mono cartridges have suspensions that allow for vertical movement of the cantilever even if such movement does not generate a signal because this acts like springs and shocks on a car—it allows the cartridge to better track vertical movements from warps and other bumps and reduces stress on the stylus and cantilever from such movement.