The GOP wants to avoid a divisive vote on extending the extra Obamacare subsidies and to prove to the world that the Democrats are a weak party with no leverage in the legislative process. The Democrats want to show that they are willing to fight for average Americans and, if possible, to win; at a minimum, they also want to raise the public’s consciousness about health care and identify the Republicans as the villains in the piece.
In the end, both parties are likely to get most of what they want. The public is going to be angry when the impact of the lost subsidies becomes clear. The GOP will continue to insist that nothing can be done until the government reopens; the Democrats, with no good options, will accept vague promises of compromise; and the GOP will be forced to tear itself apart talking about the issue, probably without delivering anything meaningful to the public.