Pretty much what @Jarolin said. From my understanding web design covers websites whilst UI design covers anything that uses a graphical user interface, this covers things like websites though but could be anything where a UI needs to be thought of.
Web Design vs UI Design … hmmm. I have a few random thoughts:
1. Web Design could apply to coding only. I think there’s a significant grey area here, though. For example, if a Graphic Designer gives you a PSD mock up of what a site is to look like, complete with images and such, some may argue that you’re not designing anything, you’re structuring. I don’t know how I feel about that, though, because there’s plenty of styling (which I would call designing) that goes into creating a beautiful site … navigation, typeface, text elements; even the structure itself could be considered a design. Then again, if all this is provided to you, how much designing are you really doing?
2. UI Design, I think, applies more to function. A website about your dog doesn’t require any UI design. A website like Facebook or Twitter, however, does.
Final thoughts: I think that UI Design and Graphic Design are in the category of “vision and dream” … “this is what I THINK it should LOOK like” or “this is what I THINK it should WORK like” where as Web Design and Web Development are more in the category of “bringing it to fruition” … “This is HOW we make it LOOK this way” and “this is HOW we make it WORK this way”.
Not sure if this was what you were looking for but it’s what came to my mind. Again, just my opinions.
It’s one of the few professions where function follows form … in most cases. For that reason I actually believe that Web Development and Web Design are WAY harder than UI Design and Graphic Design. The latter professions get a free range of creativity to work with, the prior professions are locked into making someone else’s vision work completely.