I think they are different.
When talking about "Weight & Balance", it is more about how an operator / pilot would load up the plane. E.g. how to interpret the charts in the aircraft manual.
When talking about "Center of Gravity", it is more about airplane designs, and the focus is on the theoretical side. E.g. how to place engines such that the center of thrust pass through the center of gravity.
Both are part of the daily lives of the pilots, as a very importany part of preflight preparations is doing w&b calculations to ensure the cg is within permitted limits. In GA the pilot does this, in (bigger) airlines usually dispatch handles crunching the numbers and PIC checks them as he, of course, will be solely responsible for the safe operation of the aircraft.
w&b calculations are a method to determine cg. As tags, I'd definitely say that cg is a subcategory of w&b, kind of in the same manner as fixed wing is a subcategory of aircraft.