A study by political scientist Chris Parker, per the Monkey Cage, identifies a powerful and virtually defining trait of the Tea Party: its adherents believe that President Obama is not just a liberal in the tradition of Clinton, Gore, Kennedy, etc., but a unique threat to the country:

To capture the difference between the two camps, we asked whether or not people thought the following proposition true: "Barack Obama is destroying the country." If Tea Party conservatives were as extreme as some suggest, we thought asking a question of this type, i.e., fairly extreme, would tease out differences between the two camps. The data suggest the two conservative factions are divided on this question. Under these circumstances, 6% of non-Tea Party conservatives believe the president is destroying the country versus the 71% of Tea Party conservatives who believe this to be true. 

If you want to understand why a government shutdown is very, very likely, this is the fundamental fact you need to grasp. Indeed, Republicans not only have substantive differences with the administration, they understand that their base is averse to compromise in principle and deeply suspicious that its leaders will "go Washington." Indeed, reports the Wall Street Journal, Republicans are afraid of the very idea of negotiating with Obama:

The coming days represent a crucial stage in the negotiations if both sides are to avert a government shutdown. Last week, Vice President Joe Biden played a role in keeping the talks on track after a session broke up in acrimony. But he has been trying to keep a low profile, said one Democrat familiar with the negotiations, out of concern that Republicans would rather not be seen as making a deal with the White House.
"I think there's a certain reluctance to negotiate with the White House directly," the Democrat said. "Some of that is, I think, institutional. Some of it is political."

There's a logic here: If you think that Obama is destroying the country, then any deal that Obama agrees to is bad, right? But that makes it pretty hard to make a deal.