First, I'm not writing off the Bears chances, just as I'm not suggesting that losing Schaub for the season is anything but a negative.
However, when you look at the two situations and try to form an opinion on which team is in the better situation to weather their specific circumstances, it is a very defensible position to say it's the Texans.
As has already been pointed out, their current playoff positioning is better than the Bears, and their remaining schedule is probably a bit tougher than ours as half the teams we play currently have 3 wins or fewer.
I think the most important single component of why the Texans should deal with the loss of a starting QB better is the running game. When you look at the running and passing (including sacks) percentage to this point in the season, you'll see that the Texans have run the ball approx. 54% of the time, while the Bears number stands at approx. 44%. The YPC is actually pretty comparable at 4.4 for the Texans to 4.3 for the Bears, but this stat tells me the Texans offense is less impacted by the QB than the Bears. It also tells me there's likely to be less of an adjustment to the new QB for the Texans (although the way Martz's mind works, he may choose not to make any adjustments - which would present problems in and of itself).
The final item that makes me believe the Texans are in a better position to deal with this adversity is their defense. While the OP characterized both defenses as great, it didn't point out that if you evaluate defenses by yards, the Bears are in the bottom 1/3 of the league, while the Texans are #1. If you'd rather go by points allowed, then you're looking at a middle of the pack Bears (13th) compared to the second best scoring defense in the league. The result of all this is a Bears team that has an average point differential of plus 6.1 per game compared to a Texans team of plus 10.7 game. In other words, we're better positioned to deal with a decrease in offensive potency.
Again, neither team should view it as a positive that their starting QB is out of the rest of the year, nor should either team view it as a sign that their season is over. I just think that in a head to head comparison, the impact on the Texans shouldn't be as great as the impact on the Bears.