Ogre is a tool for mainly hardcore coders/programmers. As mentioned before: it is just one part of any solution: a 3D Graphics Engine.
Which si just part of anything you want to build, and the rest you have to build yourself (or depend on other 3rd party projects).
The benefit of BGE in this is:
it is a ready out of the box complete game engine.
And it doesn’t stop here… it has an intergrated 3D system, to manage and create (almost) everything you need:
3d modeler / animations / logic bricks / physics / easy to learn (standardized Open Source!) scripting (python)
And as mentioned before: it has a very active and enthusiastic community, dedicated to improve and progress the BGE development.
The difference between Ogre and BGE is thus not so much the logic bricks or python,…
because if you are already daunted by logic bricks and python,… then you probably can forget about Ogre…
Ogre needs you to be able program on a higher level (you can do this with Blender / BGE, but you do not need to!),
you need to know how to write code, compile, and make bridges between components of 3rd party solutions.
Ogre is much more complex and requires a lot more time to get it to function like the BGE already does.
So for the most of us, if you look for a Game Engine, BGE is the logic choice, even if you are a hardcore programmer on your own.
That said, BGE is GPL, while Ogre falls under MIT license.
The difference: what you do with Ogre is up to you, where Blender falls under GPL.
WHile your own content (Blender output) is seen as DATA, thus open to your own choice of license.