"Menthol Mint" is a trade name associated with Dr. McGillicuddy's schnapps. In essence, it is a standard peppermint schnapps, but they have coined the name 'Menthol Mint' to imply that the product is different from other similar products.
The author of this article has sampled Dr. McGillicuddy's as well as the cheaper, Hiram Walker Peppermint Schnapps, and can find no difference between the two.
Peppermint Schnapps are traditionally associated with a winter-time, or Christmas beverage. It has been proposed that this is related to the alcohol having the effect of drawing blood to the surface of the skin, thus making the drinker feel warmer in a cold climate. The warming feeling is similar to that of drinking brandy, but in actuality lowers the drinker's body temperature, despite feeling warmer.
As stated before, "Menthol Mint" is more of a trademark, rather than an actual product, the actual product being peppermint schnapps; thus, a more accurate description can be found in either Peppermint

Peppermint

Peppermint is a hybrid mint, a cross between the watermint and spearmint . The plant, indigenous to Europe, is now widespread in cultivation throughout all regions of the world...

 or Schnapps

Schnapps

Schnapps is a type of distilled alcoholic beverage. The English word schnapps is derived from the German Schnaps , which can refer to any strong alcoholic drink but particularly those containing at least 32% ABV...

.