Scheme is cleaner conceptually, but that's about it.
Common Lisp is based on older Lisp concepts(on which Scheme is based too), and it takes some good ideas from Scheme(such as lexical scope), while retaining older Lisp concepts too, it also adds a cool object systems, proper error handling, a large, useful and well-thought of "library", optional dynamic scoping, a very nice condition system, fine control over the compiler/evaluator, separate function/variable namespaces, packages, real macros, a large variety of native types, key/rest/optional args, and many more. You can implement a lot of these in Scheme, but that's the problem, you HAVE to implement them, or use some libraries which are not portable across Scheme implementations, to achieve what you get out of the box in any of the CL implementations. This doesn't mean CL doesn't have this problem to a slight degree: networking, FFI and threading is not standardized, which means people have written compatibility layers to achieve this across different CL implementations, but it's not a real problem for most Lispers.
This makes CL an instrustrial-strength Lisp, making it fit for practical usage, while Scheme is a theorethically beautiful, distilled Lisp, which is liked by academics. Since I prefer writing real applications, I find Common Lisp considerably more useful.