What is the difference between embryology and developmental biology?
It's a simple sub-set relationship, embryology is a sub-set of developmental (it's a "specialty"). Embryology is only concerned with the embryo - developmental is a larger group, concerned with developments that may occur in other stages of life. Such as the development of secondary sexual characteristics in humans - a developmental thing, but not embryonic. 

These are not purely objective definitions and the dividing lines are not well defined. Labels for various scientific fields are only generally applicable, and they are very flexible. For example, many Chemists also do a lot of physics. Many Biologists do a lot of Chemistry. Many Embryology studies dip into developmental stuff. These divisions are not always clear.

Embryology is a part of Developmental Biology in the same way that Biology is part of "Science" - simple sub-set relationship. If you don't understand sub-sets very clearly, you'll want to study up on that relationship - it's very important in ALL sciences, so you can understand things like "All squirrels are animals, but not all animals are squirells"

This statement would be reasonable - "All Embrology is Developmental Biology, but some Developmental Biology is NOT Embryology"