This is quoted from something I made a note of when I was wondering the same thing. I forget where I got it but it's in my file of Maya questions and answers that I have been accumulating over the years....
Creating a Parent setup (what i usually call hParent) places the child object under the heirarchy of the parent. The child object is free to move independently of the parent.
If you set up a Constrain Parent (cParent) will establish a link between the channels of the parent to the child. You cannot keyframe it independently of the parent.
So if you want to move around the child after you parent it… i’d suggest just to parent them (press p)
"If I have seen further it is by standing on the shoulders of giants." Sir Isaac Newton, 1675