How important is the material/contact surface of a suction cup?

both surfaces have to be as smooth as possible, in the beginning of your pressing down the air under the cup still can get out, do the volume of air decreases, the vacuum or very low air pressure under the cup ist what keeps it fast.

Key to getting a suction cup to effectively stick is to have the lip of the cup made from compliant material. It will then conform to microscopic irregularities in the mating surface.

Larger irregularities will tend to locally push the cup lip up and out of contact with the mating surface over a nonvanishing distance, potentially creating a leak. Hence the cup lip must have a certain width to get a satisfactory seal with a rough or irregular surface.

Thanks to Some additional research, the answer is that when depressing the Cup, pressure under the cup actually increases and the differential pushes a lot of the air out. Once equalized this transfer ceases but after the cup recovers shape, increases volume, the air expands and The pressure drops allowing atmospheric pressure to take over and hold the suction cup Tightly to the surface. This differential must be much less than the differential that Initially pushed the air out.

A suction cup grips because atmospheric pressure pushes it against a surface, and the seal is in contact only on a small part of the cup area. So, smoothness and compliance of the seal material is important ONLY on the load-bearing part of the area; the whole 'cup' can be a crude iron hemisphere, as long as the contacting rim has a seal material which, at the vacuum of interest, doesn't leak atmosphere into the evacuated center. Such a seal is under the force of 1 atm times the area of the full cup, but only touches the flat surface on a small seal-contact area. The roughness of the surface determines the amount of pressure required to establish a secure seal: the seal average pressure equals the atmospheric pressure times the cup area, divided by the seal contact area.

This is sometimes called the principle of unsupported area, and applies to a variety of seal strategies.

Some molded suction cups are applied nearly flat, then deformed into a cone-like shape to form the vacuum and seal. If such a cup is allowed to relax to flat, it comes loose because there's no longer any unsupported (evacuated side of the cup) area.