I still think it more properly fits in the Alien section. Even if the AVP movies lay the groundwork for them, these institutions only exist further down the track.
As regards the question, I would think that Colonial Administration is a sub-division of Weyland-Yutani; an administration concerned with managing its various colonies, such as Hadley's Hope.
From what Ripley says about the ICC in Aliens, it seems to be an entirely separate entity, essentially the same as the various corporate watch-dogs we have currently. Their function would be to monitor all the various corporations in respect to ratified interstellar law.
That said, in terms of who has the greatest power, it really depends. If I'm right, the ICC would theoretically be more powerful than the corporations in monitors, simply because it has the power to hold them to the standards of the law. Of course that would depend largely on the transparency of the corporations it is there to monitor, and the autonomy of the ICC itself. In terms of wealth and sheer amount of resources I doubt that the ICC comes even close to approximating what WY has at its disposal. As such their ability to side-step the ICC would be considerable, either through corrupting the ICC or simply overwhelming their very likely scarce resources.
The fact that WY can afford defense contracts under a government military gives you some idea of their power. It would be like Murdoch hiring the US army as his own personal security agency.
