There is a difference between teaching children that society will punish law breakers, and that the Christian God will do it. That is clearly discriminatory to Jews, Muslims, Bhuddists, and Atheists and their belief systems. Public schools are state schools and this clearly violates Constitutional separation of church and state.
It seems like a distinction without a difference to me.
Before one can respond, one must know where you found said "Constitutional separation of church and state".
You might also explain how it might be differently applied to "State schools" vs. "Federal Schools" and "Private Schools".
I'm also puzzled on how explaining a religion is discriminatory to other religions and especially to those who profess no religion at all.