Recs.
Updated
SpecsUpdate
Pros

Pro Fantastic UI toolkit (scene2d)
scene2d is a 2D scene graph, scene2d.ui is a UI toolkit on top of that. It enables very complex mobile and desktop UIs.


Pro Active and helpful community
The libGDX community, in the official libGDX forum is extremely helpful and approachable for any kind of question regardless of the its quality or difficulty.
The forums themselves are a very helpful resource for any issue or guide simply by searching past posts in there.
In addition to the forums, there's also the official #libgdx IRC channel on Freenode.
Pro Free, open source & permissive license
libGDX uses the Apache License 2.0.
Not only is libGDX free and open source but also it's license gives you a lot of power over the engine. As long as you provide a copy of the license, give credit, do not hold devs liable and do not use libGDX logo in any engine forks you can do pretty much anything you want.
Pro Focusing on object pool patterns, to control memory without pointers
Unlike Unity or other engine, it allows to optimize a language that uses garbage collector when using patterns of objects you can control the use of memory without needing a language like C / C ++, getting the same speed in a more productive language.
Cons
Con Not starter friendly
Even default applications fail to load in Android. It doesn't have any documentation on errors either.
Con Documentation could be better
Other than a brief installation / getting started overview, libGDX's documentation consists of an official wiki with several incomplete pages, and automated Javadocs. The community recognizes these shortcomings, and new users are encouraged to ask for help.


Con Does not create compiled code
LibGDX runs entirely on Java and does not create executable binaries.
Con A bit difficult to use
This engine is not well put together. Is made from various free modules each with their own peculiarities. At times it feels you need to learn a couple of libraries rather than just one. Is not an engine for beginners as it requires coding. Lots of coding. You need to be intermediate to advanced in Java to develop in LibGDX.
Recommendations
Comments
Flagged Pros + Cons
Pro Gradle support
Allows easy integration with multiple IDEs, easy dependency management and provides shell support.
Pro Supports desktop, mobile and web
Supports Windows, Mac, Linux, Android, iOS and HTML5/WebGL. Being able to develop & run mobile games on the desktop is very efficient to rapidly iterate.

Con Documentation could be better
Other than a brief installation / getting started overview, libGDX's documentation consists of an official wiki with several incomplete pages, and automated Javadocs. The community recognizes these shortcomings, and new users are encouraged to ask for help.
Out of Date Pros + Cons
Con Performance issues related to Java and the JVM
libGDX is written in Java. While there are ways to build it into native binaries, its intended and supported use is with the JVM, which has performance ramifications.
Con Java not best language for gamedev
Because all professionals in gamedev and many libraries for gamedev use C++
