I play CSGO, where people are obsessed with refresh rate and fps. I have also used 240Hz and 144Hz monitors. I would say with certainty that 180 fps would not provide a benefit between 144Hz and 240Hz. In my opinion you need a game with a steady 250-300fps for 240Hz to make a difference.
I will say however that given a price difference <$80, I would get the 240Hz for future proofing over a 144Hz.