Idea for a video
To make it easier to understand ... or adding to rhyseyness's answer
The ideal transfer of heat is through direct contact between metals. However, there's no perfect flat surface, there's some microscopic deformations, holes, dents and other things on the surface of the heatsink.
The surface of the processor is also not perfectly flat, the metal lid can be slightly concave or convex depending on how it's manufactured. Also, there are engravings on that metal which introduce very tiny holes in the metal.
So if you bring the heatsink over the cpu surface, tiny particles of air would be trapped in those holes. Air is very good to carry away heat from some surface, that's why air is pushed by fans through heatsink fins, but if it's trapped somewhere it actually behaves like an insulator, restricts transfer of heat between surfaces.
A good example of this is double glass windows, where you have two panels of glass and air trapped between them. This way, the heat inside your house won't leak out as easily as with one glass panel windows.
So in order to prevent air particles from being trapped between the heatsink and the cpu surface, we apply a bit of thermal paste. Usually we apply a tiny blob of paste in the center because as the heatsink comes down, it puts pressure on the blob and it spreads in all directions towards the edges, going inside the microscopic holes and reducing the amount of air particles, pushing them out towards the edges.
For best results, it's best to have as thin as possible layer of paste, just enough to have a thin film of paste between the metals, and filling the microscopic holes in the metals.
Metal to Metal would always be better heat transfer than metal to paste to metal and thicker layer of paste just makes the transfer harder
You see videos and articles over the net where people put all kind of substances instead of thermal pastes and for short periods of time they'll work just as well, because basically they all achieve the same thing, remove the air trapped between metals and have better heat transfer properties compared to pure air. However, as those substances go through heat-cold cycles their chemical properties may change and after some time their heat transfer property could change and they could become worse at transferring heat. Pastes are made out carefully selected materials that maintain their heat transfer properties for months to years after application.
So applying paste over the fins of the heatsink would only reduce the air's ability to leach heat from the fins, it would be an extra block. With fins, you actually want moving air over fins to take heat away so it's actually even beneficial to have microscopic holes or all kinds of ridges because those only increase the area exposed to air.
Think of cpu coolers with heatpipes that have a lot of very thin aluminum "blades" surrounding the heat pipes ... it's all about having a lot of surface area from which the air moving over the surface can pull the heat and move it away.
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