For both CAP and RI, the cAMP binding domains consist of three main helices and an eight stranded beta barrel (graphic here).
In CAP, the cAMP binds in an anti conformation, while in R it binds in a syn conformation. This is thoguht to stem from the differences in the envrionment surrounding the adenine rings of cAMP. The aromatic stacking interactions in the A and B sites of R are not present in CAP. Another factor is the absence of the short helix between beta strands 6 & 7. CAP has an extra residue in the region between the strands that prevents the helix from forming. As seen in the network section, this helix is important for the interaction the phosphate of cAMP and R.
The binding of cAMP in R is ~100 times tighter than cAMP binding in CAP, the absence of the aromatic stacking interactions and the phosphate-helix interaction