Moai SDK vs Cocos2d-x and Cocos Creator

Because it uses Lua + C++.

You can use your own custom shaders and extend the language with C++. MOAI doesn't give you everything, instead it gives you the tools you need to develop any game you want.

So it is extendable with C++

Uses a limited subset of Lua

Moai SDK is used by popular developers including Double Fine for their mobile game, Middle Manager of Justice, and for their Kickstarter hit, Broken Age(formerly, Double Fine Adventure.) It was also used to create the popular mobile game Bubble Ball 2.
This proves that this engine can be used to make fully-featured and fun games.

Has options for development like Hanappe and Rapanui, providing a different way to interact with the engine.

The Moai SDK Forum is active and users will go to great extents to answer queries, post code snippets, beta test, and even purchase and give feedback on each others' games.

Although the Moai SDK is very advanced and uses complicated features, the documentation wiki thoroughly and clearly explains how to use these features, such as the Moai Cloud(a service that provides web services for your game), and multiple ways to build from source.

25% of iPhone games are made using Cocos2d-x.

This means you will not be alone in development, and will have access to a large community. You'll know you are developing for an engine that works.

Documentation includes a programming guide, API reference, video tutorials and massive reference test code project showing all functions and giving working code to the user.

A new feature since Cocos2d-x v3.1 is support for 3D models (in your 2D game), not only this but support for skeleton animations is included too! This awesome feature allows for impressive characters in your game along with easier, more fluid and realistic animations.

It supports Lua and Javascript with full feature support.
Especially with Cocos2d-JS you can develop games cross web and native, and the native solution have great performance with JS Bindings, much better than hybrid solution.

Cocos2d-x forums are active.

Cocos2d-x is not only open source but also supported by Chukong Technologies of China and USA.

Regularly updated and adding support for the latest technologies. 2014 has already seen the release of Version 3, a new Cocos Studio development toolkit (optional) and support for new technologies like skeleton animation systems Spine and Adobe DragonBone.

Cocos2d-x is C++ based engine and it has CPU advantages for most platforms because of that. It uses polygonal mesh methods for sprite rendering for using GPU advantages. (You also use quad methods for benefit CPU).

Because it is based on Pyglet.

On top of supporting pretty much all existing platforms (except consoles), Cocos Creator (Cocos's IDE) allows you to write 1 code that runs on Android, iOS, Windows, MacOS and HTML5 (not Linux though).

Through the use of SDKBox you can easily integrate 3rd party SDKs and plugins for each version of Cocos2D (Lua, C++ or JavaScript), you just choose the SDKs to integrate and SDKBox will do the rest.

For example, if you want to add a rating plugin, you use sdkbox::PluginReview::init(); and if you want to add the Vundle Ad Network SDK, you use the one packaged in SDKBox SDKBOX sdkbox::PluginVungle::init();.

It has a built-in Python interpreter that allows for easy debugging.

Cocos Creator (Cocos' IDE) comes with scene editing, UI editor, animations & particle editors and whatnot. It's also easy to use and pretty intuitive if you read the official documentation & tutorials. Way way better than the old CocoStudio.

Hundreds of video tutorials available.