SkywalkerLAX,

As you are already aware, Grand Hyatt (GH) is closer to the Kang Nam area that is more of the business area and Itaewon while the Millenium Hilton (MH) is closer to the government center and historical areas.
Seeing the map of the area, either hotel is located within 5 minute walk of a subway station, GH to line 6 at HanGangJin Station and MH to line 1 at Seoul Train Station. Depending on where you want to go, you can ride the subway to a station near the point of interest and then take a cab for the rest. Trying to ride the cab during the day in the city will take you long time and cost you a bundle. Depending on how long you plan on being there, you might want to consider getting a subway card to use so you don't have to wait in line to get a ticket everytime you ride.
I've been inside the MH but not GH so can't compare the hospitality. The MH is near a construction site (as of Nov. 08), about 300 ft in SSW direction, and you need to go next to it to get to the subway station on downhill slope.
From either hotel, you should have no problem finding places to go sightsee by subway or taxi. I don't know where you are planning to go, but if you are going to include a visit to DMZ, then try to get on the USO tour since, I think, they do a more complete tour. Also you don't have to worry about the language.
Be sure to try out some of the food at the stalls at night in various parts of the town.
Have fun.

HJ