MGRS is just a special format for describing UTM coordinates. So the simple answer is that they are the same coordinate system.
UTM coordinates are specified as Zone Number, Meters East (the easting), and Meters North (the northing).
MGRS uses a Zone, A 2 letter code to indicate a 100km square, and an easting and northing value. Then the numeric easting and northing are abbreviated. See this link for details.
The Zones are the same between the two systems, and the MGRS easting and northing values directly correspond to digits within the UTM easting and northings.