According to Webster the two seem to be interchangeable, even in the figurative sense (damp/dampen enthusiasm are both possible). "Damp" seems to be the preferred term in blast furnaces, though.

I (BE) would use "to dampen" in the sense of to make something wet.

Thanks for your answer, Robuk. One more question:
Which term would you use in the area of absorption (e.g. vibration absorption)?
Leo proposes "vibration dampening" - could you also say "vibration damping"?
If not, what would be the difference?

Personally I would say damping in that case, but Webster allows both.

Thank you for your quick answer.

Der OED hat bei "dampen" den Zusatz: (Now chiefly U.S.) - vielleicht ist das der kleine Unterschied.