The basic difference has something to do with the purpose of the design. A web designer focuses on the visual side of the website while the UX designer ensures there's great user experience with the website and its pages.

Overall I would ask - why are you asking the question? Is it because you have a large budget and you are embarking on a major website build?

If not I would not worry about hiring a UX or UI designer. The best step is to hire a web producer, rather than a designer. A designer is very good at designing, but that does not mean they are able to develop a USP, write effective coversion copy, and carry a project to completion.

The internet is unfortunately filled with millions of business websites that people think "look really nice", but do not generate any leads or enquiries.

Developing a solid USP and having a producer with you who can develop a web property that is effective in generating sales is the right path, rather than be concerned about having a UX designer on board.

Just my 2c  (:o)
 

This is a simplest explanation for the layman.
 
A web designer focuses on designing the visual elements of a website such as graphics, videos, typography, colour, etc. They can also be skilled in SEO, mobile and responsive design and user testing. They will know codes such as JavaScript, HTML and web technologies but not to the extent of a web developer. A web designer’s expertise is specialised in websites across all devices and may or may not do mobile apps. However, they usually don’t cover software and other product.
 
On the other hand, a UX designer which stands for user experience designer focuses not on visual elements, but on the user’s experience in using and interacting with all your business’s touchpoints. This covers not only websites but software, apps, prototypes and products. Their job is to make users have the most positive, seamless and meaningful experience possible. They will have expertise in user research and behaviour, prototyping, user testing, etc.
 
There’s another professional and that is a UI designer, which stands for user interface designer. Their goal is similar to a UX designer which is creating the most positive experience for the user but he or she focuses specifically on the user interface, working on any screen that a user uses, touches or interacts.  UX and UI designers tend to go hand in hand. Both are not limited to websites or web applications unlike a web designer.
 
Which one does your small business need? It depends on what your project is and the goal. There are instances when you may need all 3, especially when working on a website. 

This question made me smile! 
When you visit a website, you automatically rate it - it's fast, it's slow, it's easy to buy, it's a nightmare paying for something or it's a breeze, it looks good, it looks bad etc. It's all about the experience.

A Ux or Ui designer will tell you these things and more. But so will a good web designer/technologist - we jsut don't use Geekspeak.

I often think that a lot of terminology we have to deal with is useless geek-speak, and that's how I feel about these terms, Ux and Ui. It sounds important, but really, if your website isn't a smooth and pleasant experience then you'll know all about it and so will your visitors - without the geek-speak!

Let me enlarge on this. Color is important - very, very important for its pscyhological effects on a website visitor. And yet we don't have 'Cx designers' as a separate entity  - we expect our designers to have that expertise and to design something that is optimised for readers and buyers, that won't stress their eyes or put them off. The same is true IMO of Ux and Ui - ease of use is a non-negoriable reality.

When we (or any other competent agency) are designing a site, we put all of our experience into it - the experience and knowledge of artists, web technologists and business managers such as myself. We do not expect customers to go chasing Ui, Ux, Cx, Bx or any other x factors. 

Some people will be offended by this opinion. But by profession I'm an engineer and business manager. When i was describing high-tech products and systems to customers I stayed away from technical terms and spoke about benefits. I do the same with my own business.