Coldness is the absence of heat. When we remove heat energy from a system, we say it gets colder. It's a way of describing the reduction of molecular activity resulting in the sensation of heat. So the more heat we pull out of a system, the colder it gets. Cold is a description of a circumstance in which heat is missing.
Heat is energy which can be measured.
When you remove heat, the temperature goes down. Here's another way of looking at it. I don't mean those little round sugar-coated lumps you buy at the donut shop. I mean the hole itself. Donut holes are actually what's left when the middle is cut out of a donut.