It's kind of an outdated title in my opinion - I haven't really heard the term used regularly in like 10+ years. It's also rather vague. When it comes to building and maintaining a web site I think it tends to fall into two broader categories, those that manage the front end UI (user interface) and all the graphical work, perhaps the CSS as well and maybe some javascript. And those who take care of the back end development of the site - server side scripting such as PHP or ASP, manage the databases the site may be run off of, work with the servers, and all that stuff. There are people who can do both, which is great, but their time will be stretched rather thin if they got a lot on their plate.
I tend to think of "Web Designers" as the people handling the front end stuf, and "Web Developers" as the people handling the back end stuff.