Leading Advantage: MATE is Linux Mint's evolved fork of GNOME 2. If you liked GNOME 2, MATE will make you feel right at home. You'll find that a few names have changed, and that the menu is contained in a single window, but, in general, no other Linux desktop reproduces the GNOME 2 desktop quite as well.
Leading Disadvantage: MATE is utilitarian to the point of conservatism. For example, its panel applets are little different -- and possibly identical -- to what GNOME 2 offered a decade ago. Given that MATE is a classical desktop, couldn't it offer a little more?
Leading Advantage: Ubuntu's Unity makes far better use of screen space than other Linux desktops. The movement of the menu and launcher to the left side of the screen unsettled many, but follows logically given that widescreen monitors are wider than they are tall. Similarly, Unity overcomes the limitations of the vertical screen space by having the icons at the bottom of the launcher collapse, while minimalist triangles designate active and open apps, instead of wasting space on a taskbar.
Leading Disadvantage: In order to be used across multiple devices, Unity remains a desktop for undemanding users. Others might find it awkward, especially when they want to work from two or more windows open side by side.
Leading Advantage: Xfce bills itself as a balance between speed and user convenience. Xfce generally lives up to this billing, but it has an even a greater advantage: the ability to run both GNOME and KDE applications better than any other Linux desktop. If you prefer to choose your applications individually, instead of confining yourself to the applications designed for your system, then Xfce is a logical choice.
Leading Disadvantage: Like Cinnamon, Xfce lacks the ability to drag and drop icons that is so central to the modern desktop. While you can learn to manage without this ability, at first not having it makes Xfce feel old-fashioned and limited. This impression is reinforced by Xfce's long delays between releases, which can rival those of the Debian distribution.
Here, I have given a combination of my reactions and those that others have voiced on the Internet. However, the choice of desktops can be intensely personal -- many of us are, after all, spending 8-14 hours a day in contact with them.
So, under these circumstances, it only seems fair to ask: what are your main reasons for using or not using the main Linux desktops? Please respond in the Comments section.
Also see: Best Linux Desktop: Top 10 Candidates