The difference between web design and UI design is becoming more and more blurred everyday. For a long time web design was about creating websites that didn't look awful. Sites in the 1990s were designed mostly by computer geeks for computer geeks and no one really expected anything to look good. CSS allowed people to start doing really attractive and beautiful things with the web, and sites like http://www.csszengarden.com/
began to demonstrate the amazing things that you could do with it.
Then people went crazy with it. "Wow, what a beautiful 2GB picture, I HAVE to put that on my homepage!" "I love navy blue and neon orange, I'm going to build my ENTIRE site in those colors." "Oooo... my marketing department wants lots of data from customers so I'll just put this 500 question survey together and make it mandatory for customers to fill it out before they can check out with their cart." yada yada... and so the UI designers started to come along and define the procedures, and the visual cues, and the elements of website design that you CAN do but which you probably SHOULDN'T. And even better they started to develop and identify a lot of the emerging standards which make up best practices today. Pick up a copy of one of Steve Krug's books to get a good idea of how UI designers approach the design process (I'd recommend Don't Make Me Think!
, but his most recent is also good Rocket Surgery Made Easy
I think long term, you will see them diverge again because web design is such a huge field. As we do more interesting and more exciting things, our field becomes too broad for one person to do everything well. Designers will begin to specialize again (we're already seeing "site designers" who've master Photoshop and Illustrator, drift apart from designers who specialize in social media, and "multimedia designers" who work mostly in audio and video. But I believe the core elements of UI design will be as necessary to all designers as learning English or whatever language your site content is in.