Comment by NoboruWataya

I think this may be a translation issue, as I would always have understood "modern art" to refer to the movement that began in the 1860s, and (English) Wikipedia appears to back this up. Like others have said, "contemporary art" means art of the current time.

> I've seen a lot of contemporary art museums, but few modern art museums.

My experience is the opposite, I have seen plenty of modern art museums, most famously (probably) the Museum of Modern Art in New York. Some modern art museums also feature contemporary art but nevertheless their main focus and brand is modern art (eg, the Tate Modern).