Aside from the obvious possible difference of size, the PC monitor will generally have a much smaller dot pitch and will be capable of higher resolutions (above 31kHz which is 640x480p). The arcade monitor is probably either fixed mode or 15-31k (or there are some 31-38k floating around now, too). In general, the smaller the dot pitch, the more accentuated the details of the picture will be, and scanlines will be more obvious. 640x480 on a 29" TV tube (arcade monitor) looks quite beautiful with properly designed graphics found in arcade and console games, whereas 640x480 on even a 21" PC can look a bit jagged and "grainy".
The arcade monitor may also expect larger signals (~3.3Vpp instead of 0.7Vpp) than a PC monitor, but many/most arcade monitors capable of 31kHz+ have at least an option to accept PC level signals. Most provide a standard VGA (HD-15) plug as they were in fact designed to be plugged directly into PCs. Consider most arcade games in the very late 90s and early 2000s, when 31kHz CRT arcade monitors were most popular, were in fact PC based.