Web developers and designers are often involved in the same projects, but have very different skill sets and approaches to building websites.
Web designers think in creative terms
Most web designers who I know have been to college to study graphic design in one form or another. This includes designing for print and the web. They see building websites in terms of how it will look rather than how it will work. The will ponder over which tint to use and how many pixels something should be indented. They have the ability to create several design concepts from scratch based on a brief.
Web developers think in functional terms
In contrast, when building websites, web developers think in terms of how it will work rather than how it will look. A lot of their work maybe behind the scenes building robust, secure and scalable applications that allow you to run your website in the best way possible. They will ponder over what the most efficient way is to write some code to perform a particular task. Although by no means essential, a lot of web developers who I know have been to college to study a computer science related course.
Are some people both web developers & designers?
In short the answer is yes, but in practice this is usually not the case. Web design is such a different activity to web development that it is difficult to become specialised in both. The people who claim to be both designers and developers usually can do a bit of both, so when it comes to hiring a web designer or developer pay close attention to their CV and consider whether you want to hire a specialist or more of an all rounder.