I want to talk to you. I have finally got around to doing it. Don't jump the gun and assume too quickly that a company is not interested in you. Keep your message in front of them until they make that call. Your resume may have been lost in the mail, misplaced, accidentally thrown away, etc.
Some companies require it.
Some companies, especially larger corporations, will only deal with people who follow up. This is usually for highly desirable positions for which they get hundreds or even thousands of resumes pouring in and they need to screen most of these people out. By only dealing with the follow ups, these companies save themselves the work of reading all those resumes, and they end up with people who have demonstrated that they are truly interested in the position. You should telephone after you send a resume. But always back up the phone call with a follow up letter and another copy of the resume.