C1. "Fresh" upgrades

In a "fresh" upgrade you use the liveDVD of the new release to perform a new installation and to overwrite your existing partitions.

A "Fresh" upgrade consists of the following steps:

Making a backup of the data
Making a backup of the software selection
Performing a fresh installation using the liveDVD of the new release
Restoring the data
Restoring the software selection
This is the recommended way to upgrade Linux Mint and it offers the following advantages:

Safe: Your data is backed up externally. Whatever mistake you make or whatever bug happens during the installation cannot affect it.
Fast: The installation usually lasts 15 minutes. The DVD for the new release contains compressed data. Downloading the ISO and upgrading from the DVD is much faster than upgrading the system from the repositories.
Reliable: First, you get the opportunity to test your hardware detection in the new release using the liveDVD. If anything is wrong you can simply decide not to upgrade, it's not too late. Second, you end up with a fresh installation of Linux Mint, i.e. a system that was fully tested by the development team and the community.
Easy: Things do go as planned this way.
C2. "Package" upgrades

A "package" upgrade consists of the following steps:

Pointing APT to the repositories of the newer release
Asking APT to perform a full upgrade
APT is the package management system used by Linux Mint. Alternatively, some releases were given a graphical upgrade tool to perform these steps.

This way of upgrading Linux Mint should only be recommended to advanced users.

Here are the pros and cons of upgrading the system this way:

Cons:

Slow: APT will download the new version of all the packages installed on your system. Using a fresh upgrade you could have downloaded all that data by simply getting the ISO.
Unreliable: Depending on your modifications, your sources, your added software and your configuration you could end up with a system that acts and feels really different than a brand new version of the newer Linux Mint release. You're far from the beaten track and the added features might not work as well on your system as they were designed to.
Risky: The temptation when you upgrade with APT is not to perform backups... since your partitions aren't overwritten, nothing "forces" you to make backups... think about the risk though.
Complicated: Packages conflict with each others, they can bring complex dependencies and put you in situations that are difficult to solve.
Pros:

Automated: APT does everything for you (well, until something goes wrong of course)
Real upgrade: A "fresh" upgrade is kind of like the new Linux Mint with your data on it... this in comparison feels more like "your system" running the newer version underneath.