It is broadly true that the left motor cortex is predominantly involved in the control of the right limbs, and vice versa. This isn’t absolutely true — you still see ipsilateral (same side) activation during movement, although its role in motor control isn’t very clear. There are slight differences in this activation in left and right handed people, and usually the hemisphere opposite the dominant hand is slightly more engaged in ipsilateral movements. Again, the functional role of this difference isn’t very clear.

There are other functional differences, beyond motor control. Language is typically lateralized to the left hemisphere, but greater right hemisphere lateralization is more common in the left handed (and in women).