If so, would the acid snow just be as damaging?
Yes, there is, and it can be more of a problem than acid rain.
For one, when snow is falling it spends more time travelling through the atmosphere before hitting the ground than rain. As a result of this, it has more time to collect pollutants, and can become more acidic than typical acid rain for the same region. There's additional factors that complicate this however, such as the solubility of the pollutant, the temperature and the water content of the snow, etc. (e.g. warmer snow which is more "wet" is going to dissolve more than cooler, drier snow.)
Once the snow hits the ground, the dissolved pollutants are going to stay in solution, or trapped within the crystalline structure of the snow until it melts. Because of this, snow can pile up and then thaw, releasing all the pollutants and acidic water in a relatively short amount of time. This can have fairly substantial effects on the environment.
You can read a bit more about this here:
Acid snow is also mentioned in passing on the acid rain wikipedia page.
The "fairly substantial effects on the environment", as described by the links above, largely involve a large release of acid during the spring thaw, which effectively releases all of the acid built up over the winter in a relatively short period. This can result in lowered pH in waterways, which can affect fish, amphibians and crustaceans.