Based on Susan's recommendation, I went to Shabu Hachi on Friday. I had planned to try the Shabu Shabu, but when I got there I saw they had Chanko-nabe. It is a sort of meat (I ordered chicken) and vegetable stew, which I ordered in a Miso broth. It is also available in a soy broth. It is known as the food Sumo wrestlers eat. It was really great. I'd take it over Shabu Shabu at either place. :) Much richer in flavor. Main ingredients seemed to be chicken meatballs, fish cake, fried bean curd, lots of cabbage, seaweed, and mushrooms.
My girlfriend did have the regular chicken Shabu Shabu, which I sampled. For roughly the same price, I think Mizu is a bit cheaper despite the trendy atmosphere, Shabu Hachi provides a larger plate of meat, but a smaller plate of vegetables. Shabu Hachi has thicker slices of meat, but Mizu has a higher quality meat. Both places had a similar soy dipping sauce, but the other one (I forget what it is called) is pretty different. Shabu Hachi's is more smooth, and Mizu's is more grainy - both were great.
In summary, if I had to eat at one restaurant, I'd go to Shabu Hachi and get the Chanko-nabe. If I really wanted to get Shabu Shabu, I'd go to Mizu. More vegetables, and even the trendy atmosphere are what do it for me. :)
Anyway, thanks Susan, for the tip on Shabu Hachi!