To put it short, Facebook for personal and LinkedIn for professional affairs.
That said, Facebook doesn't offer as much as it used to. Nowadays I mainly use it only for chatting with friends. I'd delete my profile if I could easily chat with friends elsewhere. Too bad I can't so I "have" to remain in Facebook.
LinkedIn is a great tool to give you some credibility. For example, I frequently sell domains to end users. I contact them directly by email. At the end of the sales letter (which is customized yet very short), I put my signature with full name, title, company address and phone number. I believe recipients now and then google the sender's (my) name, to see what comes up. Among the top results is my LinkedIn profile. I've put a photo & education and job history of myself there. I've written what I'm currently doing. I also have listed my email
and even phone number (only for logged in users, though). I have some connections as well. This hopefully tells people I'm conducting legitimate business and not trying to scam them.
In other words, having a completed LinkedIn profile can boost your sales. People trust LinkedIn, moreover they trust a person who tells something about himself. And shows his photo. And has some (LinkedIn) connections. You obviously know yourself you are not a fraud - now show it to other people by having a great LinkedIn profile!