I'm relatively new to Traveller, only having discovered it a few months ago. As such, I don't really know that much about the Third Imperium other than that somebody made a cool map program for it. I am somewhat dimly aware that it is fairly flexible, but I have also read that Traveller fans often get rapped up in the setting's minutiae. Thus, I'll be ignoring the Third Imperium for the most part.
Instead I plan on looking at Traveller the same way I look at D&D. It provides a number of tools and materials for me to build my own setting -- likely a sandbox -- using its assumptions. The Terran Imperium will have scouts and marines and divide territory into subsectors, just like the Third Imperium. To me, this is the equivalent of having magic users and beholders in a D&D game. I do plan to change some things, it is my own setting after all, but the assumptions that undergird the two settings will be remarkably similar.
Ultimately Traveller is a system that I will use to run science fiction adventures of my design for my players. Usually these adventures will include the usual set up, with players taking the roll of freebooters, smugglers, and other "honest" businessmen, in an empire too large to be authoritarian. There may be some variance from that model, but I more or less view it the same way I do dungeon crawling in D&D. It's part of the game, and if I wanted to do something else I'd pick a different game.
The ultimate advantage of Mongoose Traveller, aside from its truly kick ass character generation rules, is its flexibility in this regard. The Imperium is only very lightly present in the core rulebook, and it clearly provides mechanisms for creating other milieus. The aliens provided are only samples, and there are a plethora of suggested FTL travel options -- though I'll likely be sticking to jump travel for now. Mongoose's version provides a framework that allows for a multitude of Science Fiction settings, as anyone can see by looking at the number of settings they offer themselves. While I'll be sticking close to the assumptions of the Third Imperium, I do not feel bound by those assumptions in any way.