All right, try this on for size:
In betrayal mode, the race that you select at the start of the game (which is normally the one that is first to be spacefaring) is instead killed by you, completely. This game mode has a completely different agenda -- all that stuff with the federation doesn't even exist. You are able to have a population of robot-citizens under your control (like any other race), and you capture planets and so forth just like any other race would.
For the most part, a lot of your robots' attributes (racial compatibility, manufacturing rates, combat strength, etc) carry over from the race you exterminated. Your job is to then exterminate the other 7 races and take over the entire solar system for just you and your mindless robot servitors.
Instead of the Federation Progress window, there is an Armada Management window that lets you pay credits or resources for armadas (either building them or upgrading them), and assign them to various duties (guarding planets or outposts, or attacking enemy planets or outposts).
In essence, the entire premise of the game (story-wise) is abandoned. This is your evil twin, so to speak.
It's cool, because so much of the game is the same between both, but your goals and how you utilize all the mechanics really differs because of the different endgame, etc. I'm pretty excited about it.
Invasion Mode is much less drastic of a difference, being a closer variant off the base game mechanics. But it also provides a pretty different flow, because you have to deal with the Obscura, a race that has obliterated one of the other races in that mode, and which is not a faction you can negotiate with. They're coming for everybody, essentially, and so that complicates matters because now you have the usual wars and whatnot, plus this whole other outside threat that is creeping in.