When comparing libGDX vs CryEngine, the Slant community recommends CryEngine for most people. In the question“What are the best 3D game engines?” CryEngine is ranked 7th while libGDX is ranked 9th. The most important reason people chose CryEngine is:
CryEngine has some C#template and also C# based system to write your function/ideas in to your game.
Specs
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Pros
Pro Lots of resources to learn from
Lots of references, tutorials and open source code to learn from.

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 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 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 Uses Box2D
A Java port of the Box2D physics engine is included in libGDX.

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 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 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 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 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 Kotlin support
It is written in Java so you can easily make games using Kotlin and Ktx project will help you get all advantages of this language.
Pro Very easy to customize
Pro Constantly improved
LibGDX itself is pretty mature, and get updates not quite frequently, but various libraries for it are actively updated.
Pro Across platform support
You can write once and run anywhere (Android, IOS, Desktop, Browser) also lately came with lib to deal with VR.
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 Lots of tutorials to get you going
Even though the official documentation might be lacking, there are many tutorials on YouTube for libGDX.
Pro Excellent font rendering support
Very good tools for rendering fonts.
Pro Similar to the Microsoft XNA framework
Old XNA users may like libGDX since it's API is actually very similar to XNA's.
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.
Pro C# integration
CryEngine has some C#template and also C# based system to write your function/ideas in to your game.
Pro DX12, Vulkan support
CryEngine 5.4 now supports DX12 and Vulkan
Pro Features allowing for realistic weather effects
Cry Engine has volumetric fog rendering which allows for realistic cloud shadows that actually render shadows onto the fog itself. Combined with their time of day system, it's possible to create incredibly realistic weather effects. On top of this, color grading allows user to post process pallets allowing them to change the color tone for different type weather, such as using a deep dark blue for rain.
Pro There is already a built-in AI
It can shoot, walk on patrol points, can see the player and so on.
Pro VR support
Cryengine (starting on Cryengine V) has Virtual Reality support. Developers can create games with VR support for multiple platforms: PlayStation VR, Oculus Rift and HTC Vive.
Pro Dynamic water rendering
Cry Engine has realistic water effects that even simulate ocean physics. Features such as waves that respond to global wind, and dynamic water volume tessellation allow for some of the most realistic water effects available to a game developer. The engine also takes into account LOD (level of detail) on water geometry to allow it to stay performant for water at a distance.
Pro Realistic rendering of vegetation and landscapes
Where Cry Engine really shines is with rendering scenes of nature. The Crysis games feature incredibly detailed vegetation and weather effects and it's the Cry Engine that enables that. The engine has many features to create a cohesive realistic looking world. Dynamic water effects allow users to have beautiful oceans, fog and cloud effects allow for realistic weather, and a plethora of lighting effects optimized for natural looking scenes make Cry Engine one of the best engines for creating vast beautiful landscapes.
By having all these features built together from the ground up, Cry Engine is capable of doing more complex effects more efficiently, than other engines that didn't have these effects planned from their inception.
Pro All platforms, including next-gen consoles, are supported
Supports miltiple platforms including: Xbox One, Xbox 360, PS3, PS4, Wii U, PC, iOS, and Android.
Pro Versatile flow diagram script model
Flow graphs resemble flowcharts where each box represents a function or value, with connections between them representing program flow. This provides a better at-a-glance indication of game logic than a simple list of events, and makes complex behaviors easier to accomplish.
Pro Dedicated channel for Q&A
Crytek has launched a dedicated Q&A forum for everything Cryengine related. It's called Cryengine Answers and it's a community dedicated to sharing and answering any question related to Cryengine.
Pro Online marketplace available
The Cryengine marketplace is an online marketplace which enables developers to access and use individual assets from thousands of materials, sounds and 3D objects created by the community. Even Crytek's own library assets are available there.
Pro Advanced volumetric cloud system
Cryengine has an optimized volumetric cloud system for Virtual Reality to give clouds full 3D spatial rendering. This ensures a high rendering quality with a minimal performance hit.
Pro Terrain tools are great
Pro Disallows bad practices in asset creation
Simply by looking at the RC log when exporting can greatly improve your work. Cryengine doesn't handhold you constantly and helps greatly with avoiding bad practices in asset creation.
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 The project slowed down in development
The number of active developers has decreased, many open issues and pull requests.
Con Build system is clunky
LibGDX uses Gradle, which is very demanding of memory and makes IDEs freeze on anything if your computer is not powerful enough.

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.
Con Steep learning curve
Except for basic FPS games getting anything done will require solid knowledge of C++, Flash, ActionScript and Lua.
Con No GNU/Linux support
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Con Hard to develop games other than FPS
Cryengine is a great engine to be used for developing an FPS (and it's relatively easy to do so). But if you want to develop another type of game, it requires at least advanced knowledge of C++ and Visual Studio.
Con No Mac OS X support
Con Restrictive license
Cryengine is not restrictive anymore just more personalized. The model is Pay what you want and if you want more you get a membership with them. Or private support, help and lessons directly from the CryEngine team.
