What is the connection between pedagogy and education?

In essence, education is the result of pedagogy. 
Pedagogy is the art and science of teaching. It involves not just knowledge of subject matter, but the specialized knowledge of how to communicate that subject matter to learners. Sometimes a concept needs to be broken down into parts and then put back together for and with students. Sometimes students need to approach a skill through trying it with the teacher and then trying it by themselves. Sometimes students need multiple, different explanations of a single concept. A good pedagogue discerns what students need in order to be able to learn and then figures out how to provide that. 
While the word "education" can be used in ways that appear to cover "pedagogy," a main focus of education is to change people. A person begins school unable to read or do arithmetic and ends high school with all kinds of skills, knowledge, and abilities. This is education. People who are an important part of making education happen, teachers, often called "educators," are essentially pedagogues.