Well for me, there could be quite a bit of difference between a Christian and a Child of God.

CHILD OF GOD

Definitely more inclusive from a religious perspective, but even beyond that, a child of God can include everyone who has lived, is living and will live on the face of the Earth, religious or otherwise. Obviously, God has to be real, and our relationship with Him has to be Father to child, but all that being true, everyone is involved.

Even if you take away religions who don't believe in a fatherly God, or ones that don't believe in a corporeal being, you still catch a wider spectrum of folks than you would Christian.

Now, there may be a specific religion that I'm not a where of like "The Children of God", but other than something being that specific, and fairly limiting, the term Child of God widens out quite a bit.

CHRISTIAN

Can be pretty confining, limited to only those churches that fall specifically under Christian, or it could incorporate all churches that teach Christ is the Son of God, our Savior and Redeemer. Personally, that's a better definition of a Christian, since the rest involves the interpreting of doctrine/scripture, along with whatever additional ritual/ceremony, etc., that is performed or not performed, etc.

At any rate, Christian is more of specific faith (though still huge), while Child of God could literally encompass us all.