ASDF appears to have been deprecated in favor of Quicklisp, or rather, Quicklisp appears to be the next generation of ASDF, since I saw asdf functions inside of Quicklisp.
No, ASDF and Quicklisp do different things. ASDF handles compiling and loading systems, whereas Quicklisp handles downloading systems and their dependencies. ASDF is kind of like make, and Quicklisp is kind of like a package manager.