It's political theater, is all it is. The Republican Party has been dealt a fantastically shitty hand, and they're playing it as best as they can.
The Republican Party unquestionably lost the last election, even though the political makeup of both chambers stayed relatively the same. They should have made big gains- facing an embattled incumbent with low approval ratings during a seemingly endless recession. Historically, that's a recipe for an opposition win. If they want to mitigate the damage, they have to look strong in opposition to the Democrats, while admitting the realities of the last election. If the Democrats are able to lay the blame on House Republicans again for a 'no-nothing' Congress, they are going to get slaughtered in the 2014 elections.
Meanwhile, Democrats are in the position where more people would blame the Republicans for a failure to reach a compromise. (Extensive polling gives a 45-35 margin, the other 10 being the undecideds.) Politics isn't a zero-sum game in a two-party system; both parties can lose, so long as one loses less. It is in Democratic interests to go past this fake "cliff", and write legislation after the fact rescinding any increases. Furthermore, if taxes have already gone up, they will have more leverage when it comes to getting the tax increases that they want.
TL;DR version- Republicans lose far more than Democrats by allowing tax increases and budget cuts to occur; hence, the Democrats have most of the leverage in making a deal. However, Republicans still have to appear strong.
This was the logic behind Boehner's "Plan B". (Seriously- what was he thinking? How many times have Republicans already voted against Plan B?) It was a deal unpalatable to enough of his caucus so that he wouldn't be able to get it through the House, but it was an attempt to offer a deal that would make it look like he would reason with President Obama (and Senate Democrats), just at a different level. Look strong, and force your opponent to look unreasonable. It didn't work because the tactic wasn't refused by the Democrats, but ignored. Boehner didn't have the votes for it, and they knew it.
There's no pressing need for a last-minute deal. The Four Horsemen don't leap from the clouds if there's not a new deal in place by January 1st. So you won't see one. The Democrats have no need to make a deal right now.
So think about it farther. The Democrats have no need to make a deal right now; where does that leave a smart Republican planner? With the failure of "Plan B", he has two options. First, accepting a prior deal is off of the table, because Republicans have less leverage now than they did before the Plan B bluff was called. If Republicans want a deal by January 1st to avoid being blamed (more than Democrats), they now have to do it on entirely Democratic terms. Second, you take the lumps you get from the "fiscal cliff", and hope that the bleeding that the Democrats take is enough to bring them to the table on better terms than you'd receive now.
It's a very lousy hand for the Republicans to play, but it's what they have. I expect Boehner to take the second option, and we'll see a deal in late January. It'll be easier to recover from a fake "fiscal cliff" and the 45-35 blame they'll take, versus the appearance of having completely caved to Democrats in negotiations.