>Guile-Emacs loads, compiles and executes the programs included in Emacs by default, plus Gnus, Org-Mode, and more.
Common Lisp is a language, while what they are integrating to Emacs isn't a language, it is a VM used by an implementation of a language. The VM itself is generic enough to be compatible with Elisp (although I believe there was some struggle to deal with the differing way nil and #t work between Elisp and the Guile VM).
It is a bit like how both Java and Ruby can run on the JVM, Java being born with it and Ruby being ported to it through JRuby.
A particular implementation of Common Lisp can have a VM that is similarly compatible with Emacs, although all GNU Common Lisp implementations are not as actively maintained as Guile is.
On the language side, Common Lisp is definitely incompatible with Elisp, as they have differing semantics. Historically, Common Lisp has had lexical binding, while Elisp has dynamic binding. (From Emacs 24 on, one can turn on lexical binding on a file by file basis.) Elisp also doesn't have closures. Finally, the numeric tower is different between the two.