>>>What's the difference, from a theological perspective, between "Before Christ" and "Before the Common Era" when BOTH are based on the same event reference?
Because the word Christ has very specific meaning theologically, The Anointed One. The Greek word Xristos has been used since pre-Christian times to translate the Hebrew word Messiah. Jews do not acknowledge Jesus to be the Messiah. BCE can acknowledge a common historical figure without acknowledging divinity.
I was going to say how about Before Jesus then, but thought better of the abbreviation.