For anyone who doesn't know - a big difference that Common Lisp brings to the table is that it has separate namespaces for variables and functions. This may not sound like much but it makes it much easier to write macros (code that writes code) and facilitates compile-time computing.
Another big difference between Scheme and Common Lisp is that the Common Lisp specification is _much_ larger and it comes with a much larger collection of standard built-in functions and macros and a standard object-oriented programming facility, among many other things.
I know this because I've spent the last six years implementing Clasp (github.com/clasp-developers/clasp) - a Common Lisp that interoperates with C++ and uses LLVM as the backend. There are several excellent implementations of Common Lisp - Steel Bank Common Lisp is a great place to start.