[KyleGoetz]

飲みたくありません sounds more formal to me than 飲みたくないです, but I've never had much of an ear for subtle formality, so I could be wrong. It's likely just because everyone I know informally uses 飲みたくない, so that sense of plain-ness bleeds over in my perception of 飲みたくないです, while 飲みたくありません has no connotation to me.