Web artist is basically and mostly a web designer who visualise and represent his/her idea in the form of unique web experience. Not all web designers are web artists.
Normally when you say web designer it means a person who can design websites and has knowledge of using web design tools/applications, which doesn’t necessarily means that he/she has a proper design sense.
Graphic Designer and Website Designer has always been under misconception by non-design people who are looking for design services and one window solution. Due to heavy competition in this industry most Graphic Designers pretend that they know Website Design as well just for the sake of business which end up as a non user friendly experience of website. In some cases flaws in website user experience can be a result of client’s feedback those who are less-aware of Design Aesthetics.