Great question many times these topics are used interchangeably , and there is a big difference between a web designer and a web developer.
Lets start with the absolute basics.
A web designer does exactly as the name implies, they design the appearance of web pages and websites. They normally come up with a color palette, unless you are working with an established business who will normally have standards colors in use for the business. Sometimes, actually most times, very specific colors if the business is long established and has registered trademarks. A good web designer will have a good understanding of what types of sites are trending on the web and how user engagement is measured on a web site to unsure their design is meeting the expectations of the customer. Normally a trained web designer will have a background in art, computer graphics, color palettes and how marketing trends can be used in design. A good career but the pay is based on the artistic value of the designer. This can be very subjective and large fluctuations is compensation are noted in this industry. Very few good web designers become good web developers as they are two very different personalities and completely different skills sets. A good web designer will have a very basic understanding as to how web pages are laid out to keep their designs realistic and achievable and will collaborate with web developers to understand the newest changes in the programming technologies to better incorporate those changes in their designs.
So in a nutshell, what do you need to know be a web developer in the basic sense....
3. PHP - (or some server side programming language) - PHP is the most popular
5. Database language - MySQL - (or some other DB language) - MySQL is the most popular
6. An Understanding of SEO (Search Engine Optimization)
This in no way this is all that is needed to be on the upper end of the pay scales. Most of us, myself included, are constantly adding to our skill sets and learning new technologies all the time. This is a very dynamic field, with updates occurring constantly as technology changes. One thing to keep in mind is that once you learn one programming language and learn it well, most all languages are very similar, so it is not like learning all over again as you pick up new skills. As an example all programming languages, use Arrays, Variables, Conditional Statements, Functions, error handling, etc. Once you understand how these work in one language you understand how they work in all languages. The only difference is that they may be structured slightly different. They will work the same.
One of the biggest difference between web developers and web designers is that web developers can be taught. All the skills necessary can be learned. That is not necessarily true for web designers. Web designers will have a knack or built in ability to understand art, to be appreciative of color, to mix images and text in such a way as to be visually appealing. In my opinion those are not learned traits, some people are just born with artistic abilities that most of us do not have.
One other thing to be aware of web developers can design web sites, they may not be the most visually appealing, and the colors may not be perfect, they may not look as good as they could or should but I see it all the time. My personal sites and my training sites I have designed myself and believe me I am not a web designer by any stretch. They are functional and serve their purpose. But when I am working on customer sites, I always employ a professional web designer. They are well worth their cost when it comes to designing professional sites.
As you decide to make your decision as to what direction you want to take in this field, stay away from plug and play web site developers like WordPress, Joomla, Dreamweaver, or any of the out of the box build a website in 30 minute kits. These will do nothing to add to your skills other than teach you very bad habits in web development, and also show a complete lack of understanding on your part of site security and SEO to any future employers. As none of these tools, and I will prove that to anyone that challenges this statement, follows Best Practices, site security measures, or proper SEO. It is not the fault of the tools, like WordPress, Joomla, Dreamweaver, etc, as in some ways these tools can be quite useful for people developing their own sites for personal use that do not have a concern for security, Best Practices of SEO, it is not possible with the size and scope of these tools to keep up with the changes in the technology. The changes happen to quickly.
Sorry this is long but you asked a complex question and I wanted to give you a well thought out response and not a flippant response to a complex question.
This is a very lucrative field both in monetary terms and job satisfaction. I have been writing code since the 1980's and I love it. It can be challenging and frustrating at times but finding that solution to that tiny bit of code that has been driving you crazy for a week has a feeling that is hard to describe.
Where do I get these opinions from, I teach both web developers and web designers at a University level and I have been doing it for about the last 6 years as I have wound down my web development business and no longer recruit new customers I just support my existing customer base. I also running a training site on-line to teach web development languages.
I hope this helps you understand the differences in the two fields and helps you decide what direction you would like to take. Good luck it is a hell of a journey but what a ride it has been. The changes you will see in the technologies over the next 20 years will far exceed what I have seen in the past 40.
again sorry for the length of the reply, but you asked.....