What is the advantage of using a real arcade monitor for the "old school" games?

The main reason is that the real games used these same monitors!  The first arcade gane cabinets were equipped with a very basic video card, which was constrained in resolution for two reasons; first, most available early monitors were based on TV designs and so they had the same frequencies.  Second, memory was expensive and higher resolutions require much more RAM.  The simplicity produces sharply-defined but low resolution graphics.

For an emulator to reproduce this exactly, the beam of the monitor must have a one-to-one correlation with emulators graphics memory.  We can make the VGA card behave as a 100% emulation of the game board, however, now comes the problem.  To do this, the number of vertical lines and horizontal pixels that the VGA card displays on the monitor must be exactly the same as the original game board.

If they are not the exact same, we have to re-sample the original memory mapping and stretch or compress it to fit the screen, or put up with an incorrectly sized picture.  So the 100% emulation is lost if hardware stretching or a scan converter is used.  So a game which was originally designed to run at, say, 320 X 240 must run the monitor at exactly the same resolution to give a 100% emulation.  If it is run at any other resolution, just because the monitor is capable of handling it, the quality of the original game will be compromised, even if the resolution is higher.