I just had my first private-sector interview in almost 40 years - a phone interview for an independent contractor position. Was an interesting experience, and a positive one. I sent a follow up e-mail an hour later, thanking the manager, re-stating my commitment and experience, and wishing her and the organization well.
Since my main career was with the federal government, my experience with this subject is probably not typical. Half of my interviews were with the manager of my department, for promotions within. I already worked side-by-side with him every day, and we all pretty much knew which candidate would be selected, so other than a thank-you at the end of the interview, I never bothered with an e-mail.
In all of the career development sessions we had, I don't recall anyone suggesting a follow-up email. The emphasis was on the in-person interaction: greeting the interviewer and shaking their hand, then having a solid closing statement, and shaking their hand and thanking them for the opportunity before leaving.
However, if I wasn't selected for job after interviewing with a different department, I always asked for a follow-up meeting to review my application and interview, and get feedback on what I might have done differently, or what I would need to do to make a better candidate. That way, they knew you were serious and thinking long-term, and that might influence them to drop your name to other managers.