Are there significant differences between the platform versions of Mass Effect 3?

The Xbox 360 version is different in that it features Kinect support. Here's an excerpt from the Destructoid review detailing the Kinect functionality:

Using Microsoft's camera accessory with the game is totally optional and serves to allow most of the game's basic commands to be given by voice. Kinect can be used to issue commands to squad members, activate special abilities, and change weapons, removing the need to open the game's power wheel, pausing the game and interrupting the flow of combat. Spoken instructions also extend to making dialog choices, opening doors, examining objects and pretty much anything else one would just press the 'A' button to do in a passive manner.

The marketing push for Mass Effect 3 states that the game is "better with Kinect," and it's hard not to agree. Voice recognition is very accurate, usually able to pick up commands even when speech is rushed. It's not perfect and does occasionally, rarely, fail to recognize a command but the system proves impressive nonetheless and it's easy to completely dispense with the power wheel altogether after giving it a little test.

There are no significant gameplay differences between the versions.

Be aware that some pre-order DLC will only be available to certain users based on purchase location or ME3 merchandise purchases.

If you own the additional Kinect peripheral for the 360, there is enabled voice recognition for combat. But the game itself is not changed from version to version.