Web design: how it looks and feels
Web design is the visual and experiential side of a website — layout, colour, typography, imagery, and how a visitor moves through the page toward an action. A good designer thinks about the decision the visitor is making and structures the page around it. Design decides what the site says and how it feels; it doesn't make the site function on its own.
Web development: how it works
Web development is the building — turning the design into a working site with code. Front-end development builds what users see and interact with (the buttons, forms, and animations); back-end development handles the servers, databases, and logic behind them. Development decides whether the site is fast, secure, and actually does what the design promised.
Why you usually need both
A beautiful design that's badly built is slow and breaks; a well-built site with no design thinking converts poorly. The best websites come from design and development working together from the start, not design handing a picture to development at the end. Some people ("full-stack" designers/developers) do both; on larger builds they're separate specialists who collaborate.