There are differences in cart loading and capacitance; usually fixed for MM and variable for MC, but you heard the biggest issue, increased noise due to more gain.
Putting MM cart through MC phono stage
Once I forgot to switch the phono stage from MC to MM when changing cartridge respectively. At first I was wondering why on earth the volume must be turned only to about 9-10 o'clock whereas before it had to be turned to about 1 or 2 o'clock to get optimum sound. When I realised the mistake I listened it both in MC and MM mode and did not find much difference save for slightly higher hiss level in MM mode when the apm volumu had to be turned high. Anyone has experience with this issue? My guess is that listening a MM cart through MC phono stage might prolong the life of tubes since they are run on lower volume level. Or is this entirely stupid idea?