A year ago, Mate hadn't even reached general release. However, since then, it has been influential in making Linux Mint the distribution of choice among experienced users. Similarly, after years of being the third most popular desktop environment, Xfce has become one of the major alternatives.
However, despite their similarities, which one is likely to appeal to you depends on what you are looking for in an interface.
The newfound popularity of the two desktops is explained largely by the fact that both are based on GNOME 2. Mate is a fork of GNOME 2, openly intended as an alternative for those dissatisfied with GNOME 3 and Ubuntu's Unity. Having spent several years consciously imitating it, Xfce also resembles GNOME 2, down to the wording of many menu items and dialog boxes.
As a result, both Mate and Xfce can be classified as traditional desktops. They consist of a desktop display, a panel and a launcher, and both are largely free of 3-D effects, the influence of mobile devices or any effort to innovate in any major design elements. Contrariwise, each includes the technically useless screen-saver, presumably in keeping with tradition and users' expectations.
Each is an obvious example of the type of interface that was introduced in the mid-1990s, and remained dominant until the last four years.
However, the aims of Mate and Xfce differ strongly, to judge from each project's home page. Mate proclaims itself "the traditional desktop environment," and so far its developers have sought to do little else than to continue to make a popular desktop environment easily available.
By contrast, Xfce's home page summarizes the desktop environment as "fast and low on system resources, while still being visually appealing and user friendly." Like Mate, Xfce generally lives up to its self-description.
In addition, Xfce still shows signs of its more geeky past. The project only added user-friendliness as its goal around 2006. Even a few years after that, Xfce hadn't completed the simple, user-friendly task of adding a desktop launcher.
Instead, for the first year of its existence, Xfce emphasized speed and a small footprint. Evidence of these priorities lingers even now in the interface's inconsistencies and the tendency towards buttons and lists in dialogs and configuration settings. One or two dialogs, such as the Setting Editor might still seem formidable to less experienced users. So might the file manager's option to "Open Parents."
Nor has Xfce spent as much time as modern alternatives like Unity have in worrying about such details as rounded-corners or the width of scroll and slider bars. Despite the introduction of user-friendliness as an equal goal, to this day Xfce tends to have a blocky, slightly awkward appearance.
By contrast, Mate draws upon a decade of incremental development for GNOME 2. If it is less consistent and less current than modern desktop environments, it is still more consistent and less old-fashioned in appearance than Xfce.
At the same time, if Mate is faster than GNOME 3, it is less responsive than Xfce in every way imaginable, from start time to the speed with which windows open and shut.
But these generalities are only part of the story. There may be individual features scattered through both desktop environments that influence your choice as well.