Al Capone, at least, was not the just a mob boss but a genius at PR and local politics, (to this day he is still well-regarded by many in Cicero). He was not typical of Mafiosa but he was not unusual in the higher echelons. The Klan were mostly backwoods farmers, even at the higher levels, who made up in violence and hate what they lacked in intelligence and sophistication.
Additionally the Mafia was known and recognized as a criminal organization (though an approved one because its main business was flouting a law most people disapproved of anyway.) The Klan, at least ostensibly, sold themselves as moralistic reformers.
The most powerful Klan was brought down by a sex scandal. The Mafia would have little problem exploiting this weakness. They would set up the Klan members with hookers and alcohol and then send in their personal cops and reporters.