Constraints

There are 5 main types of constraints : Point (only mimics Translate attribute), Aim (where child aims at parent),Orient (only mimics Rotation), Scale (mimics uniform scale change) and Parent (mimics Translate and Rotate attributes). Unlike in Parenting in Constraints the first selected object becomes Parent, and also children can't move independently.

Unlike the parenting, constraints can be toggled on and off , it can be animated.

Parenting

If you want to group things, parenting is the way to go. It is important thought to remember which object is parent (last selected object becomes parent after hitting 'P' key) because 'children' can move independently from parent, but when you start to move parent object all children objects will follow it. 

(to unparent hit shift+P)

Like you can see here, door is a parent for glass, when I move the door glass goes with it, but if I move glass,door stays on it's place.