Well, the most important difference between Frontier: Elite 2 and Elite is flight mechanics.
Elite was an arcade space sim/trading game. Frontier, despite maintaining general style (and relative simplicity) goes for realism here and is Newtonian, so you get to experience more or less actual spaceflight, though it's still much easier than the real thing, since your ship has lots of fuel, powerful engines (allowing you to disregard stuff like transfer orbits) and is very effectively stabilized. Oh, and you get an autopilot that makes for very effective 'training wheels' (too effective actually, as most commanders seem to have never taken them off), although it can be suicidally stupid on rare occasions.
Frontier is effectively a hardcore game with very casual learning curve, with the only hurdle being having to get used to the whole "no friction in space" thing and "everything is relative" which make huge difference in combat (along with "engines off" misnomer that only turns off computer controls and gives you direct control over thrusters which is absolutely crucial in deep space combat) and when you're flying several thousand km/s and need several days of full burn to stop (which will be a big problem if you only start braking when you can actually see the planet as more than a point of light).
Frontier is also bigger, there are around a billion of star systems, laid out as proper (albeit flattened and with reduced systems count - sort of a slice) galaxy, planetary systems are realistically scaled and animated (which means that to land on a planet or dock on station you first need to catch up with it), and everything is relative (which means that "how fast am I going" is not a valid question unless you specify your frame of reference).
And then, you get multiple ships to chose from, can take up all types of contracts and jobs via BBS, ability to mine planets and asteroids and to scoop hydrogen fuel from gas giants and stars and so on.
Elite might have been the most influential space sim in history, but it's the Frontier (and properly patched FFE which is effectively Frontier++) that I'd consider the best and most ambitious one.