We used marble flooring throughout the house. One advantage of stone over ceramic is that stone can be precisely cut, whereas ceramic tiles can't be. This let us place the tiles close together (appx 1/16"), to minimize the amount of grout. This was important to us because grout is hard to keep clean. Using larger 18" tiles allowed further minimization of the grout.
We chose a white, plain grout material, and did not apply a sealant afterwards. The marble itself was also not sealed.
Here is a picture of the floor in our living room and dining room:
We chose stone rather than hardwood for several reasons. First, it was the most compatible with our choice of a slab foundation. A hardwood floor would have to have been installed on top of wooden runners. The runners would have been nailed to the concrete, and the floor would then have been nailed to the runners. This would have left a 1/2" air gap under the floor, which would have been a perfect place for mold to grow. After we moved in, we had a water spill from our washing machine -- this would have been a huge problem with a wood floor. Another problem with wood is that it has to be finished, and the finishing materials are in general fairly toxic. They also need to be refinished every 10 years or so, with heavy sanding, lots of dust, etc. Stone floors avoided all of those problems.
We chose marble rather than other stones because of its attractive appearance. The cost was roughly the same as a high-quality hardwood floor, at appx. $10/square foot, including materials and installation.