I do not think that there is an explicit answer to this question anywhere in canon, at least that I know of. The only thing that it can be compared to is the dominant hand (as Himarm mentioned in the comments) in our muggle world. If you are a right handed person, you can learn, rather easily, how to throw a ball consistently with your right hand. However, if you were to try to throw the ball with your left hand, the mechanics are likely to be off (comparable to wand movements [swish and flick]) and the resulting throw is likely to be sloppy, not and off target.
Being off target, along with improper wand motions, is likely going to hinder the caster of a spell as they are not as likely to hit their target, or to create the proper wand motion necessary for the spell. This significantly affects the ability of the caster to be effective with their spell work, but does not necessarily affect their ability to cast a spell.
All of that said, I cannot imagine why a witch/wizard would not be able to, with practice, learn how to use their non-dominant hand with enough proficiency to be able to comfortably, and accurately cast using either hand. This would be especially true of those witches or wizards who may have lost their dominant arm to dark magic, as Moody did with his leg.