The basic conclusion I came to after initially starting this thread, doing a bunch of math, and spending a lot of thought on it is that they are fairly close in power.
If you had to put one above the other, it would be Battle Master. Battle Master also wins hands down if you have a lot of short rests.
What Champion needs to come out on top with damage dealing is lots of attack rolls--so advantage
really helps here, as does Great Weapon Master if you include feats. In fact, at very high levels Champion with Great Weapon Master out-damages Battle Master by default. The initiative bonus that Champion gets also means that they will get more attacks overall than a similarly statted Battle Master, which synergized with their abilities well.
If you aren't really interested in managing a Battle Master's maneuvers, you don't have to feel like you are getting the shaft with Champion. You'll have more athletic prowess, better initiative, likely an extra +1 AC eventually, and you shine when short rests aren't available or when you can gain advantage on attacks (so get your allies to use things like
faerie fire and watch yourself dominate the fight).
I'm actually planning on playing a Champion straight out of basic, default human, not even using feats. I want to go with someone with a simple fighting style and great athletic prowess (it's a savage setting, and I envision him leaping across a chasm swinging some mighty weapon at a foe mid-jump) and that's the way to do it.
This really hasn't been my experience with 5e so far... but whatever.
EDIT: I also find it interesting that on these forums we have some posters claiming that the casters have been nerfed too far with concentration limitations... while others are claiming they are akin to gods in the game... which one is it?
My guess is that they relate to a couple different play styles. Sure, casters can be over-contributers if you take advantage of all of the tactical possibilities, both in and out of combat. Scouring the internet helps in this area.
On the other hand, if you just pick up the Player's Handbook, come up with an interesting character concept and some reasonably good sounding tactics, they aren't going to overshadow anyone.
Or in other words, while you can break anything with heavy optimization, casters can apparently be broken in more and more interesting ways. I have no personal experience with this in my groups.