A web designer is a person who designs the front end of the website. "The Part You See on your screen"
Take for example everything we see on this website right now. Someone who designed the looks of this website would be a web designer.
A web developer codes the back end of the website "The Part We Can't See" the developer is coding and developing the part that makes the website actually work.
The web developer works with more complex coding to actually make a website work and do what it's supposed to do.