You assume the heatsink comes with a thermal pad already on it. Many don't, because basically they've got horrible heat transfer properties, just barely good enough to work.
I hope "clip one side of first...then push the other side it" means "clip the side away from the socket cam box (the flat part that the lever rotates in), level the heatsink onto the processor gently, then push it flat, then hook the other side of the clip without letting the heatsink tilt or twist". 🙂 If you do it any other way, you can damage the processor or the heatsink. The "push it flat" part is the most important.