I'm really just going by reviews here, but the FX01 sounds like the better camera to me. While it's noise at higher ISO setting normally gets mentioned, it seems to get high marks for almost everything else, from it's lens, autofocus, exposure, metering, color, and other aspects of image quality to performance, handling, menus, battery life, and LCD. And Panasonic is known to have the best image stabilization system in their cameras.
The V570 seems to get more average reviews. Alot of people love having the 23mm lens, Kodak always gets good marks for ease of use, and they recently won a JD power award for customer satisfaction as well. But while the V570 seems comparable in ease of use, menus, battery life and LCD, it may fall a bit short of the FX01 in performance and image quality, particularly in indoor flash shots with the wide angle lens (it has a weaker flash and not quite as good lens quality).
But if you really want the wider angle, the Kodak is a good choice. 23mm is noticably wider than 28mm, and alot of realtors prefer the 570 for that reason, for photographing interiors. For all-around use, I'd go with the FX01.