I did some math. Since the thread has kind of died I didn't bother bringing my results, but since it's started up, let me share.
My gut feeling was correct. All other things being equal, the Battle Master out damages the Champion and there is no way around it (other than gross mismanagement of superiority dice by the Battle Master).
I compared the Battle Master and Champion expected damage output at levels 6, 11, 15, and 20. My baseline assumption is the standard human variant, single class, since we are looking at it from the perspective of the simple option. I also went with a greatsword, although it doesn't appear to matter as long as both characters use the same weapon. I assume that a Battle Master is using maneuvers that add his superiority die to damage as well as whatever other effect is included.
At level 6 it would take the Champion 18 rounds of combat between short rests to catch up to the damage output of a Battle Master.
At level 11 that becomes 22 rounds.
At level 15 the Battle Master's ability to regain a superiority die if he is out at the beginning of combat means that the Champion cannot catch up in damage output unless you are dealing with extremely long individual combats (in the vicinity of 13 rounds per battle).
At level 20 the Champion reaches his maximum potential and can equal the Battle Master for damage output as long as there are 13 rounds of combat between short rests (not per battle as at level 15). This seems pretty reasonable, but it requires level 20 to get there.
If you break the principle of going simple and allow both characters to take the Great Weapon Master feat, things look a lot better for the Champion.
At both 6th level and 11th level a GWM Champion can equal Battle Master (GWM or not) in damage output with 8 rounds of combat between short rests.
At 15th level the GWM Champion only needs 4 rounds to catch up.
At 20th level the GWM Champion is caught up in 3 only three rounds.
So if you allow the Great Weapon Master feat the Champion is at least equal at low-level and straight up wins at higher level. Otherwise the Champion is far weaker in damage output, and only becomes potentially competitive at very high levels.
Now, this assume all things are equal. There are three things (and from what I can tell, pretty much only three things) that improve the damage output of the Champion relative the Battle Master.
1. Additional attacks
2. Improved chance to crit on attacks (such as from Advantage)
3. Improved damage on a crit (such as rolling additional damage dice on a crit only--probably through multiclassing)
Nothing else appears to make any significant difference.
So if your fighter is hasted or otherwise granted additional attacks through magic or otherwise, or if he is frequently getting advantage on attacks, Champion will catch up and at some point (I haven't even tried to figure out when that would be, as it is dependent on the nature of the actual effects) even exceed the damage output of the Battle Master. Given a campaign with a normal level of magic and other ways of gaining advantage, it is possible this may even out the damage output of the classes, although in general, my gut tells me that Battle Master still comes out ahead.
Now, I'm a strong critic of the theory that class balance should be based on DPR. There are a lot more things that go into what a class is and what they can do. I'm taking DPR into account because it is an objective measure of comparison. For that comparison to be meaningful we also have to look at the other features that separate the archetypes. For simplicity's sake, I'll mostly assume we are looking at the finished characters at level 20, which is most likely to favor the Champion.
So other than damage output, here is how the Champion and Battle Master compare.
Champion
-Bonus to non-proficient physical ability checks. Since most Champions will be proficient in Athletics, I'll assume it won't apply there. This means it applies to Acrobatics, Sleight of Hand, Stealth, and any other Str, Dex, and Con ability checks that aren't dependent on skills.
-Bonus to jumping distance.
-Bonus to initiative checks.
-Extra Fighting Style. This can allow a degree of flexibility, but for the simplest possible comparison I'll assume that the additional style is Defense for a +1 AC.
-Nice regeneration ability.
Battle Master
-Free artisan tool proficiency.
-Ability to evaluate opponent.
-A variety of special effects through maneuvers.
I'm going to make some personal comparisons here, though others might disagree.
-Bonuses to ability checks and jumping distance are roughly equivalent to the ability to evaluate an opponent, and artisan's tools can be learned with downtime and so have no real relevance. These abilities can be considered to cancel each other out as a matter of preference and playstyle.
-The remaining non-DPR features seem to me to favor Battle Master slightly, but others may disagree.
Conclusions
This leaves us with the balance considerations being that, assuming a campaign goes to high-level, the Champion gets a +1 AC, half proficiency bonus to initiative checks and a nice high-level regeneration ability, compared to Battle Master gaining a variety of special effects through his maneuvers and generally higher damage output (all things being equal).
My personal feeling is that that still puts Battle Master ahead. However if we assume that magic and advantage in a typical D&D campaign is sufficient to even out the damage output, and that we are taking our characters to high-level, then I think they are pretty well balanced.
In other thoughts, I find it interesting that most people seem to focus on the warlord aspects of the Battle Master, while what I really like about them is the maneuvers that have no effect on their allies but are all about granting them cool tricks like disarming, pushing, and otherwise messing with their opponents. I like to think of the subclass as ideal for gladiators and swashbucklers.