A really good web designer has to be a mixture of a talented artist/designer and a coder/developer.
If the web designer is mainly a developer, knows his code, etc. but doesn't have any artistic talents, then you end up with one of those millions of web sites, which are functional, possibly clean, but otherwise not very visually stimulating. In short, there is nothing there for the heart and soul. That something that instantly fills you with a feeling as you enter a website, or intrigues you to look further.
This can be good or bad, as transmitting the mentioned feeling is not always a success. In that sense the coder, as long as they keep things simple and clean, will at least not mess up, and you will have a nice, functional web site without any visual distractions, but also without any charm.
If, on the other hand, the web designer is mainly a graphic artist, but has no real coding knowledge and therefore very possibly uses a WYSIWYG ("What You See Is What You Get") editor to create web sites, you might end up with something even worse: a messy dirty code.
Clean and valid code is very important. Visual web page editors add a lot of unnecessary code, and basically generate bad markup, which is in long term, harmful to the website. Anyone who uses such an editor shouldn't call themselves a web designer. You may be an artist, you may be a designer, but in order to have the right to call yourself a web designer you should write your own code. That means typing it manually, keeping the code clean, the file size of the page as small as possible, and the usability of the web page higher.
Now consider the proper and often extensive usage of css (cascading style sheets) and all the fun that comes with experimentation, keeping up with the ever changing standards, mastering new programs, and I'm not even going to start talking about animation and its importance as part of web design.
But, when everything is finished, you are left with this unique feeling of joy and pride knowing you have single-handedly given birth to a beautiful and complex new web site.
Of course, in a large design and development company a lot of these tasks and challenges are split amongst several people or even teams, rather than being mastered by one original person, but really, where is the fun in that?