I am using CS and have imported Altium Designer libraries so I think they are compatible/same.
I believe the schematic capture between the two works pretty much the same but CS uses the Ribbon interface rather than dropdown menus. I have had no problem loading Designer Schematic files into CS. You can save Altium Designer PCBs as a version 5 and they load into CS from what I understand but I have not had much experience with this. I also understand that Designer can load in CS files but this needs to be verified. I think someone also said the shortcut keys are all different between the two so you might end up getting a bit confused between the two if you are using them both all the time but depends really how much you intend to use them both at the same time
My gut feeling is that once you master CS, using Designer is probably going to be a familiar but more feature packed experience.