There probably isn't wrong with your current SMTP setup. WordPress uses a particular method of sending out emails and this may be falling foul of the authentication and security settings that your mail server uses. This happens a lot with Google Apps and Office 365 as I mentioned, but quite often you'll find that emails that WordPress sends out end up as junk or not delivered at all because they don't appear to have come from an authorised source.
Plug-ins like WP-SMTP stop WordPress from using its usual method of sending email and starts using SMTP to send emails just like your own normal email programs do. You tell it the correct mail servers, user names and passwords and it acts like any other email client from then on. If you're using Google Apps you might get an alert about it the first time asking you to confirm that it's ok for your website to send emails in this way but after that it's plain sailing.