Today the terms website designer and website developer are often used interchangeably. But the two roles require very different skills.
A website designer is responsible for the “front end”, the look and feel of your website, how elements are arranged on the webpage and how they relate to another. His/her role requires skills similar to that of a traditional graphic/print designer, and uses similar tools (for example Adobe Photoshop, Illustrator and InDesign). Though technical skills are required to use these tools, the role itself is not a technical one.
In reality, many professionals fulfil both of these roles. A website developer may have good insight into the design process, or a website designer may learn how to code to a degree that he/she can create a functional product. Usually one skillset is stronger than the other, but as long as he/she can work competently at both, there is no reason they can’t perform a dual role.