One downside is that it's impossible to get a 12-tone scale where *all* major triads have frequency ratios 1, 5/4, 3/2.  So something has got to break when we try to switch keys and stay in just intonation.  This video by the wonderful Elam Rotem explains what breaks:

https://www.youtube.com/watch?v=XhY_7LT8eTw

(2/2)