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.

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.