What is the difference between climate footprint and CO₂?

What you measure is climate footprint and the unit with which you measure it is kg CO₂e, or only kg CO₂ if all the emissions are carbon dioxide (CO₂).

Think about it as you would with length, where what you measure is the length and the unit you measure it with is a meter. If it makes grammatical sense to write length you write climate footprint and if it makes grammatical sense to write meter then you write kg CO₂e.