gravity - Why do Earth and moon move apart but binary black holes move closer?

Here is how the tides move the moon away from the Earth:

The moon orbits the earth, and there is a difference in gravitational force between the the side of the Earth nearest the moon, and the side far from the moon.

This difference in force tends to pull the Earth into a oval shape with its long axis pointing towards the moon.

But the Earth is also spinning, and this spinning moves the axis of the oval forward, so the oval doesn't point towards the moon, but a little ahead of it. So there is a bump on the Earth, and it is permanently a little in front of the moon. This bump has mass and it pulls the moon towards it, So the moon is being pulled forwards. The moon pulls the Earth back. So the Earth's spin is slowed, but the moon gains energy and moves a little further from the Earth.

The moon has also been slowed down to the extent that the same side always faces Earth, and it will remain locked like this.

Gravitational waves, on the other hand, imply an emmision of energy from the binary system, and as energy is lost, the black holes spiral in.

There are no tides on black holes because there is nothing there to be pulled into an oval shape. The event horizon is not a solid surface. The mass of a black hole is concentrated entirely at the singularity, there is no structure to be deformed into a bump.