What I believe this book provides that CLRS does not is a series of succinct descriptions of common algorithms AND brief explanations of where you would most likely find those algorithms to be near optimal solutions.
Skiena's book does not provide a detailed theoretical analysis of each algorithm like CLRS does, nor does it include particularly exotic data structures and algorithms you'd find in journal papers. What Skiena does provide along with each algorithm is a series of references which do provide a lot of this additional background information that you could independently find if you need a more detailed look at a topic.
As a bonus the book also includes a series of "War Stories" describing how some of the algorithms solved specific real world problems. They're an amusing break from the rest of the book if you're trying to read it cover to cover rather than just as a reference.