Capture One simply processes the raw file, just as Adobe Camera Raw does. Once processed, the image can then be retouched in Photoshop.
The biggest difference between Capture One and Adobe Camera Raw is how they reference the captured color. Capture One uses ICC Camera profiles, and Phase One provides many profiles for a wide variety of light conditions for each back. It is imperative you use the best one suited to your lighting conditions to get the best color rendition. ACR uses Adobe's DNG Camera Profile in conjunction with Kodak's ProPhoto ICC profile to render color.
The two methods are very different in their approach to rendering color, but the difference hinges on how to best determine the color of light at the time of capture. ICC camera profiles require a known value to a white point (i.e., the correct color of the light source) in order to make an accurate camera profile. There's some great explanations of it on these boards, and on Adobe's forums.
If you want to try ACR with your Phase back raw images, simply convert the raw files to the DNG format and ACR will manipulate & process them using its rendering engine.