Damper is only a noun. It refers to (1) something that deadens, restrains, or depresses, (2) an adjustable plate for controlling a draft, and (3) one that deadens vibrations.1 Damper‘s corresponding verb is dampen, which means to deaden, restrain, or depress. Of course, dampen also means to make slightly wet.2
A dampener is someone or something that dampens.3 So damper and dampener can both refer to one that deadens sound vibrations. But damper does not work for something that makes something slightly wet.
Examples
Damper is only a noun. Sometimes it refers to something that deadens, restrains, or depresses. In this sense, damper is often used metaphorically, especially in the phrase put a damper on, which means discourage, diminish, or suppress. These writers use damper in these senses:
Futures were actually surging overnight before Spain’s lackluster debt auction put a damper on the proceedings. [Forbes]
But it was not the columnist that put a damper on his political hopes. [Telegraph]
And damper also refers to an adjustable plate for controlling a draft, and someone or something that deadens vibrations—for example:
Leave clear written instructions on how to use it, including opening the damper before building the fire and using a fireplace screen. [Wall Street Journal]
In comfort damper mode, the suspension provides a very agreeable ride, soaking up ruts in the road better than some classy sedans. [New Zealand Herald]
Dampen is damper‘s corresponding verb, but it also means to make slightly wet—for example:
The entry of those two lower-priced tablets will dampen Apple’s growth somewhat. [Mashable]
To dampen both [vibration and noise from the engines], Crystal has added new engine configuration. [Vancouver Sun]
A cold front that is threatening to dampen Cup Day racegoers is speeding up. [Taranaki Daily News]
And a dampener is someone or something that dampens. It’s usually used where damper would work just as well—for example:
The top tier all around the Wembley stands are empty … putting something of a dampener on proceedings. [Guardian]
Cut-price, bulk-buying mecca Costco has refused to put a dampener on reports they will be opening a store in Brisbane. [Sydney Morning Herald]
References
1. http://www.oed.com/view/Entry/47092 ^
2. http://www.thefreedictionary.com/dampen ^
3. http://www.oed.com/view/Entry/47091 ^