I would recommend using a marble threshold in the doorway between the two floors. You could install it flush with the old tile while being 1/4" higher than the new tile. Marble thresholds come in different varietes such as natural white, cultured (beige or white), and travertine and are beveled on the edges, making a nice transition from one surface to another.

Hmm....I never thought about raising the floor to meet the other one. They are two different kinds of tile though, so I would still need some way to transition between the two. I think I will probably take Michael's advice as it sounds like the easiest option. Thanks for your opinions.