Programming is the craft/art skill of manipulating digital code and data. Within programming you can learn about specific domains - for web apps that would mean databases, networking, a host of standard formats and protocols, and user interfaces. Many commercial programmers go their entire career focused only within one or two domains.
Computer science is the academic study of computation. As the rigorous curricula of today would have it, it is a more mathy endeavor than programming by itself. In particular there is a focus on provability and formality that doesn't come up in commercial programming.
Basically, stuff you learn in CS won't help you ship the ideal mix of features, stability, and time cost for (insert app here). What it will tell you is about a variety of methods with which you may approach the underlying computation problems and direct your own original research efforts.