Well, yes and no. There is the basic UTM grid, which can use either datum. Then there are derivatives of UTM, notably the Military Grid Reference System (MGRS). Just to confuse things there is also the US National Grid (USNG), which some organizations encourage the use of. As an example of how these look, here is a waypoint in a park in Anchorage. First using NAD83/WGS84:

Lat/Lon= N 61 deg 12.047 min W 149 deg 56.712 min

UTM= 06V 341750 6788716

MGRS= 06V UN 41750 88716

USNG= 06V UN 41750 88716

The main difference between UTM and MGRS is that the MGRS substitutes a two letter code for the 100,000 meter grid square, which means that in MGRS coordinates Easting and Northing are always the same number of digits. Also, I've given the MGRS to 1 meter. Typically, they would be truncated to 10 meter or 100 meters, which just shortens the coordinate. Also note that for NAD83/WGS84 the MGRS and USNG are identical.

Giving the same location in NAD27:

Lat/Lon= N 61 deg 12.014 min W 149 deg 56.396 min

UTM= 06V 341866 6788571

MGRS= 06V UC 42029 88443

USNG= 06V UN 41866 88571

Note that the military uses a different two letter code for the 100,000 meter square when using NAD27, so the datum is explicitly encoded in the coordinate. USNG uses the same two letter code as it does in WGS84. Therefore with Lat Long, UTM, and USNG, one must specify explicitly which datum is being used. Also note that my understanding is that the military now uses WGS84 exclusively for all their mapping. Only older legacy military maps would have NAD27 and the alternate 100K grid square designator.

UTM and derivatives have many handy features. A nice square metric grid, "read right up", etc. However, some of these advantages don't work so well when one has to deal with areas that straddle two UTM zones (the 6 in "06V" above refers to the UTM zone). One of my pet peeves (besides not specifying datums) is navigation book authors who sing the praises of UTM, but never talk about issues associated with zone boundaries. But that's a topic for another long post!

I'm sure most of us know that for navigation purposes, there is no accuracy difference between NAD27, and NAD83. In the example above, all the coordinates refer to exactly the same spot on the ground. As long as you know which one you are using, you can choose whichever you want.

The only time datums are an issue is when you think a coordinate is in one datum, but it really is in another. Hikermor could give a coordinate in any datum he chooses, and I could find it, if I know the datum he used. However, if hikermor told me the priceless artifact was buried at N 61 deg 12.014 min W 149 deg 56.396 min (in NAD27) but I thought he meant NAD83, I would be digging in the wrong spot. If you take a waypoint with your GPS, you can just change the datum in your settings, and the coordinate will change appropriately. But if you manually enter the waypoint with the wrong datum, it will stay wrong even if you change the datum in your GPS settings.