+ there's detail within the detail and the details in prints appear somewhat richer as a result
+ ability to crop more and still make large prints
- any camera shake is exacerbated - it takes very little movement to be seen
- optical problems with lenses will be more visible, or you may find problems you never knew you had
I wouldn't consider these "downsides", per se. It's like getting a new eyeglass prescription.
Your old eyeglasses may seem to work fine, but when you get a new and much stronger prescription, suddenly a lot of details will pop out. Specs of dust on your furniture that were invisible pop into view. The facial aberrations (minute scars, blemishes, hairs) you see in the mirror suddenly increase in great number. When you cut off other cars while driving, the other driver no longer just waves, you can now see he is using a specific hand gesture.
In the same way, the pixel resolution is like eyelgass prescriptions. With your old "12MP" prescription, the large pixels blurred fine detail so much that you couldn't see the blur caused by camera shake or the optical aberrations in your gear.
With your new "21 MP" prescription, the pixels are no longer blurring out the fine detail that obscured problems before. Now you can see the camera shake and aberrations that were always there, but hidden. Of course, you don't have to see them if you don't want to: downsize the image to the same spatial frequency (resolution) as your old camera and they will disappear. This is like putting on your old glasses.
A better solution is to improve your gear and technique enough to take greater advantage of the higher resolution.