I chose Other because if you’re asking this question then you probably haven’t even gone to the LWJGL and jME websites to read up on them.
Do that first.
As zammbi said, LWJGL (LightWeight Java Game Library) is an OpenGL binding (among other things, like OpenAL and game controller reading), whereas jME (Java Monkey Engine) includes a scenegraph, model loading, etc. jME uses LWJGL (or JOGL) as its backbone, then iterates on it to do things that are beyond the scope of LWJGL (as intended).
On a simplistic level, you basically asked,
“I want to get from one place to another. Which car should I use? This sedan or these wheels, engine, and transmission?”
But anyway. Go with jME. The point of it is to save you time. If you use LWJGL you’re going to need to learn how to code OpenGL and a you’ll need to write model loaders, texture loaders, and more. Also look into other engines as zammbi mentioned.