In a situation where your predecessor is on the skids and you're the potential replacement, hard copy is more likely to be detected than an e-mailed note to the boss. Chances are the firm's recruiter and hiring authority prefer e-mail and have become so accustomed to a digital presentation that they won't miss a handsome letter on bond paper. DO use e-mail when you are over age 40 to underscore a contemporary image. Be friendly and personable but stick to business. Omit the joking or amusing remarks that crack up your online friends.
Paper works better with traditionalists.
They like the fact that you went to more trouble to print out your letter, buy a stamp and mail it. Paper stands a better chance of being saved as a reminder of your savvy manners. Keep your letter short. Use double spacing to distinguish paragraphs. Emphasize ideas with capital letters.