I've read both books, and I think that they're aimed towards different audiences. CLRS is aimed towards computer scientists. It offers a fairly rigorous treatment of algorithms, mathematically proving correctness and performance. The Algorithm Design Manual, on the other hand, is aimed at programmers who are willing to take claims of performance and correctness at face value if it leaves room to discuss further optimization and compositions of algorithms into solutions for problems.
EDIT: Though, that's not to say I wholly endorse CLRS. I think Knuth's The Art Of Computer Programming is more detailed and more mathematically rigorous than CLRS. Unfortunately TAoCP is multiple volumes, while CLRS manages to be a single (substantial) volume.
EDIT: Though, that's not to say I wholly endorse CLRS. I think Knuth's The Art Of Computer Programming is more detailed and more mathematically rigorous than CLRS. Unfortunately TAoCP is multiple volumes, while CLRS manages to be a single (substantial) volume.