Edit: I've always done my Ubuntu upgrades using dist-upgrade.
Ubuntu doesn't upgrade like that either, or didn't in the past. You still have to change the sources for dist-upgrade to move up to the next one. People on Ubuntu 14.04.3 still use dist-upgrade to upgrade their daily or weekly patches and it doesn't move them to current releases. It takes other changes, like the sed lines here, to make Ubuntu do that.
If that is the case, what is the difference between apt-get upgrade and apt-get dist-upgrade?
Edit: I just checked and the command I've been using is do-release-upgrade... not apt-get dist-upgrade...
Yes, Ubuntu has an extra command that includes the sed stuff specifically for Ubuntu. In Mint, the base is remaining the same Ubuntu now so that command can't be used. They'd have to make their own, which they need to.
dist-upgrade includes things like the kernel that upgrade does not. There is a really old thread about it around here somewhere where I had the same questions and @thanksajdotcom was explaining to me why I needed to switch me standard patching to dist-upgrade because important patches would be missed otherwise.