What is the difference between a video game developer and a video game designer?
If you meant by developer is programmer, I'd really like to correct that.
Developer means any person that has a role in the development of the game. Consists of roles: Producer, Programmer, Artist, and Sound are the most basic high-level position in video game development. "A game developer is anyone who has any involvement with the creation of the game at all. Engineers, animators, modelers, musicians, writers, producers and designers who work on games are all game developers." (qtd. from The Art of Game Design by Jesse Schell)
If you really meant programmer/engineer when you said developer, then they are the ones who "program" the game; as in, putting all the pieces of the game - ideas, arts, sound, implementation, etc - into a whole playable state.
Designer, on the other hand, are the ones who "designs" the features in the game. Not strictly into just ideas, but rather, much more in-depth than that. Designers focus on giving details about how game features are going to be implemented. This doesn't mean they are the only ones who are limited to doing that, but it is the position where it's their sole role to think through things more in-depth. Typically, design just means "decisions about how the game should be" (qtd. Jesse Schell). But in reality says it's much more stricter than that if you mean the job position "game designer." Game Designers tend to have the most demanding wide variety of skills because you have to consider as many things as possible, including but not limited to business, animation, mathematics, music, communication, technical writing, psychology, and much more. It doesn't mean that you have to have all those skills, but the more knowledge you know the better.
A game developer is a person who can make a game from scratch by themselves, very difficult and hard to find actual qualified game developers. Development teams consist of game developers, artists (2d,3d,sound) and management level (producers, distribution managers).
A game designer is typically a game developer who has proven his/her worth in a team, has a vast amount of experience and at least one shipped title to their name. They are chosen to make executive decisions that effect the direction of development, thus why they need to be able to program, design & manage - so they can make calculated decisions. (when this is an individual it would be considered an indie developer and not a designer as designer is not an entry-level position in the games industry).
Hope this helps