The Democrats are in control of both chambers of Congress as well as the White House. This bill is a compromise - Republicans got some of what they wanted and some of what they did not want, Democrats got some of what they wanted and some of what they did not want. It has nothing to do with saving face, it's just dealing with the reality of government.
Had the Pubbies waited until January they could have gotten more of what they want and less of what they do not want, but the American people would have been struck with hikes in the tax rates in the mean time, and Democrats would remain the party with more power. Republican leadership saw this compromise as best for the country and good enough for themselves.
The right wants votes by running on a crazy right-wing ideology that sounds good to some nuts, but is terrible policy.
So they have it both ways - since Reagan, they have talked the talk to get votes of 'small government', while spending big for political benefactors for that support.
And now, they hold things like unemployment that's great for the economy, or even Reagan-started START policy for nuclear safety, hostage for tax cuts for the rich - but it's not as if they don't want to both get the vote of the nuts who like to see them have radical positions, while really wanting these things - that have overwhelming public support - passed so they aren't destroyed by them in the election.
Imagine the mother of a child telling the father, "if you don't let me win on which movie we see, I'll kidnap and withhold food from our child until you do!"
It's not as if she really wants the child to starve - she's just immoral and reckless in being willing to do so to win a battle, just like Republicans.
Suddenly, the Democrats have to have basic things for the good of the country be something that counts for 'their agenda' in negotiations.
So, the mother 'concedes' for their child to be fed as a concession to the father, in exchange for concessions from him. It's crazy, but that's how Republicans are.