Solar 2D (formerly Corona SDK) vs CopperCube

Corona SDK is completely free. That includes pro-tier plugins.

Corona SDK ships with Corona Simulator, which runs your game/app directly on your PC/Mac and updates every time you make changes.
It provides immediate feedback to your actions, you can see your changes right on the screen, without necessity to make build to device. Getting instant feedback really boosts tenfold prototyping and development speed.

It's easy to create a game that looks good on many different sized mobile devices.

With the live build feature, once you have created a build and installed on a device, you get lightning fast turnaround times because any change on the code or data is updated to the devices running the game (within the local WiFi) immediately. So changes can be tested on the real hardware within a very few seconds.
What's even more impressive, this even works flawless with multiple devices running the game. You have to use it to learn how good of a feature this is while development and even more, while doing QA. Imagine fixing bugs and everyone of your QA team/friends/whoever helps to get your game done, has all changes on his device without doing anything but waiting 5 seconds - outstanding.

Uses the great and easy-to-learn Lua programming language.

It's very quick to get things up and running with Corona SDK. The API is extensive and while it's not 100% feature-complete with the iOS API, it's close enough that you could create tons of games and never run into a roadblock.

The API docs can be found here.

Corona does not throw photoshop-like madness full of buttons editor. You can go as fast as you want, learning and building game from ground up. Eventually, you'll learn how much corona is doing for you. But to start you don't have to master complex editor software. It's a great tool to learn to start game development if you want to learn how to program and make games. Your experience will be 100% transferable to any other Pro game engine.

Corona works on OS X, Windows and Android (including Kindle Fire & Nook).

Since Corona SDK became Solar2D, it's completely free, as only some third-party plugins are paid.

Since Corona SDK became Solar2D, it's completely open source under MIT license.

Includes easy to use 3D editor for quickly clicking together 3D games.

Can create 3D games as real WebGL apps, running inside websites. Doesn't use a cross compiler, so WebGL apps are small and download quickly.

CopperCube includes a terrain editor. Terrain can be drawn with height painting tools directly in the editor, textures can be painted quickly with automatic texture blending into the terrain. There are also tools for placing grass and bushes, and for distributing meshes automatically over the terrain.

It was also written by the founder of irrlicht, although it is not open source.

CopperCube is available on Steam It was Greenlit.

You can quickly develop an experimental working model of the product (prototype), because the engine gives you access to a lot of prefabs, plugins and settings. And, you can use the visual programming to speed up the process, even if later you have to write code in order to improve the final product.

Supports both DK2 and DK1.