moving coil or moving magnetic input.

Other than impedances, it is easier for a higher gain input to be overloaded than a lower gain circuit. As such, using something other than a low output cartridge in a high gain MC circuit can produce harder, grainier output on peaks. Better phono stages have very high margins of error though, allowing a very high peak input level prior to driving them into momentary saturation. This is more likely to become a problem on recordings that have a very high / loud average recording level or recordings that have great changes in amplitude. Other than that, the differences in loading from the MM to the MC inputs may provide VERY different tonal and transient presentations, so keep that in mind. Sean
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What he said.

Actually, yes. In fact, usually, both inputs will use the same circuit with the mc adding another gain stage to the mm circuit.
Gain is important as far as the phono stage is concerned because if there is not enough, the line stage has to do more work, and there is typically more noise in the line stage circuit. For a preamp with a given noise floor, the more of a phono signal is given it the better the ratio of noise and signal you have.
On the other hand, if you are only going through one gain stage in the phono section the sound might be better than going through two.
MM cartridges are sensitive to capacitance, so the mm input will always have an input capacitance in a certain range, and mc inputs might have a different resistance because many mc's prefer something other than the 47k input that mm's are usually designed for.
One may sound better than the other, depending on the cartridge, phono and preamp. There is no rule that you must use the mc input, or that it will nesesarily sound better, you should let your ears decide.