At first, users might wonder why Linux Mint offers both its Ubuntu-based Linux Mint Standard Edition and the Linux Mint Debian Edition (LMDE). Since Debian, Linux Mint, and Ubuntu all derive from the Debian repositories, isn't the family resemblance too close to bother?
To the casual user, the choice may appear to be no more than the result of Linux Mint trying to accommodate as many users as possible. However, depending on your needs and preferences, you may find that one edition suits your needs more than the other.
In general, the two editions have much in common. Both the Debian and the standard editions are available in 32- and 64-bit downloads that default to the Cinnamon or Mate desktops. Both use the same installer, and both open for the first time on desktops with similar wallpapers and tools. Both, too, can add other desktop environments from the Mint repositories that they both share. According to Linux Mint, LMDE is faster than the standard edition, but in practice the difference is slight enough that many users probably never notice.
However, look closer, and the differences start to appear -- although these difference have changed over the years. For example, it is no longer true that the LMDE is a rolling release -- one that adds new packages as they become available, rather than waiting for a general release -- although LMDE 1 was.
Also, contrary to a widely circulating story, LMDE 2 is fully capable of using Ubuntu PPA repositories for packages in development. The PPAs simply have to be added as a package source in /etc/apt/. Alternatively, their packages can be downloaded and installed using the dpkg command. Since Debian and Ubuntu have been different distros for well over a decade now, you may find that some packages from PPAs are not compatible with Debian, but these cases are relatively rare, particularly if you stick to productivity applications rather than core system components.
The same is true of proprietary drivers or utilities like Flash. True, Debian itself installs by default with no access to these items, since its sources.list file only includes main, the free-licensed package section for repositiories. However, since LMDE 2 lists repositories with the contrib and non-free sections enabled by default, users will have no trouble downloading proprietary packages.
The main difference between editions is that while the standard edition may have a desktop application for some features, to get the same features in LMDE, users may sometimes need to alter a file in a text editor. In that sense, LMDE is less polished than the standard edition, just as Debian tends to be less polished on first bootup than Ubuntu.
However, even that is not necessarily a drawback. LMDE has just enough of Debian's Do-It-Yourself philosophy that it can lure users to learn more about administering their system, while the standard edition may encourage users more to remain on the desktop.
Debian and Ubuntu Written Small
To say that LMDE is essentially Debian and that the standard edition is essentially Ubuntu would be an over-simplification. Both are modified by Linux Mint.
For instance, unlike Debian, the current LMDE 2 has yet to migrate to Systemd. In addition, it still uses the Jessie repositories while Debian has moved on to Stretch. Although LMDE receives regular bug and security fixes, as well as backports, that lag may mean that LMDE is not as current in some places as Debian itself.
Similarly, while Debian itself lacks packages for Ubuntu's former desktop Unity, LMDE can install Unity from Mint's own repositories.
All the same, to describe the two Mint editions as Debian and Ubuntu in a microcosm does have a certain amount of truth. LMDE is based on on the Debian Stable repository -- or, to be exact, on Old Stable Jessie, currently.
Packages that have reached Debian Stable have been through extensive testing, and conform to the detailed Debian Policy Guide about how they are written and interact with the rest of a Debian system. Some Stable packages may still require bug or security fixes from time to time, but, on the whole, the results are generally the most reliable and secure packages available in Linux.
These characteristics explain why Debian has become the upstream source not only for dozens of derivative distros, but is often the distro of choice for security-focused distributions in particular. To a large extent, LMDE has similar benefits, although in theory modifications to integrate packages into Linux Mint might sometimes make it less reliable or secure than actual Debian.
By contrast, Linux Mint's standard edition depends on the latest Long Term Support release from Ubuntu. Currently, that dependency is the Xenial Xarus release. Like all Ubuntu releases, Xenial is based on packages from Debian's Testing and -- in places -- Unstable repository. As their names imply, Debian Testing and Unstable have yet to meet the high standards demanded by Stable.
By the standards of many other distributions, Testing and Unstable packages may be reliable enough, but they are less well-tested than their equivalents in Stable -- although both Ubuntu and Mint do their own testing as well. However, the advantage is that packages that arrive in Mint via Ubuntu are often more current versions than those that come directly from Debian Stable. Whether they are or not, though, depends on where Debian and Ubuntu are in their life-cycles, since the two distributions are not in sync.
To a large degree, these differences between Debian and Ubuntu carry over to Linux Mint. If your priority is stability and security, then LMDE is probably the Mint edition you should consider. For a server, for instance, LMDE would be the natural choice.
However, if you are not running a server, and would prefer to have more recent packages, then Linux Mint's standard edition is likely to suit you better. Despite updates, some packages on LMDE could be several years out of date by the time the next release comes out.
Such obsolescence could be a serious handicap, especially if you are running new hardware. For instance, the latest Linux kernel in LMDE is 3.16, while the latest in the standard edition is 4.8. Depending on your hardware, LMDE might not be able to take full advantage of a newer CPU's features unless you take the risk of adding a kernel from Debian Testing or Unstable repository.
In the end, choosing between Linux Mint Debian Edition and Mint's standard edition comes down to what matters to you. The differences are a matter of degree, and, either way, you will have made a reliable choice.