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Pro Great performance across devices
While it may not always have the best performance for a particular device or system, it's one of the few engines which consistently performs very well across different platforms.
Pro Lots of resources to learn from
Lots of references, tutorials and open source code to learn from.
Pro Allows you to migrate from 2D to 3D within same framework
LibGDX supports both 2D and 3D game development. Allowing developers to migrate from 2D to 3D with ease without having to switch the engine they are using or having to learn a new API.
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 libGDX is 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 Scene2d support
Pro Can use any JVM language
Since it's built with Java and runs on the JVM, any language that compiles to Java bytecode can be used to develop games with libGDX.
Pro Doesn't force a specific design
Unlike some engines, which only allow you to make your game in a few specific ways, libGDX allows you to design any type of game you wish as it is extremely customisable.
Pro Very easy to customize
Pro Interfacing with platform specific code
Sometimes it is necessary to access platform specific APIs, e.g., adding advertisement services or leaderboard functionality provided by frameworks.
Pro Uses Box2D
A Java port of the Box2D physics engine is included in libGDX.
Pro Constantly improved
Pro Support for 3rd party tools
libGDX has built in support for many 3rd party tools, including (but not limited to) Bullet Physics, Box2DLights and the well-praised Tiled Map Editor.
Pro Lot's of tutorials to get you going
Even though the official documentation might be lacking, there are many tutorials on YouTube for libGDX.
Pro Similar to the Microsoft XNA framework
Old XNA users may like libGDX since it's API is actually very similar to XNA's.
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.
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 Not starter friendly
Even default applications fail to load in Android. It doesn't have any documentation on errors either.
Con Does not create compiled code
LibGDX runs entirely on Java and does not create executable binaries.
Con Java not best language for gamedev
Because all professionals in gamedev and many libraries for gamedev use C++