How would you explain the difference between a programmer and developer to a layman? 

I would say, may not exactly be the right answer, but should sure work. Here it goes:

When you are new to the job, you are given guidelines on what to work on. You are free to explore on few things being given out with some guidelines and rules (That’s when you can say you are a developer).

Eventually with time when you know how things work, you add up your own customized work routines to get the job done in faster and better way. (You have been a programmer now).