I would say that one of the most important aspects of a government's response to a disaster is how it makes the victims of the disaster feel. Does it make them feel like the government cares about them? Or does it make them feel abandoned? To me, this was the major problem with the response.
The government's response to Katrina really made it look as if they did not care about the residents of New Orleans. This was made worse by the fact that the residents were black and poor and the government was Republican (and these two don't usually get along). Because of that, there seemed to be a logical explanation as to why the government wouldn't care.
Finally, there was the President praising the man whose organization was botching the job. That seemed to indicate that the government didn't really care what was happening to the victims.
You might get some different answers to this question. One answer that you probably won't find is that the government acted well and responsibly during the crisis. I would say that one fundamental failure of the government was the lack of investment in the infrastructure of New Orleans. It had been known for some time that the levees which were to protect the city in the event of cresting waters were subject to breach. All forms of government closed their eyes to this and when the walls did give way, New Orleans was submerged. Another note of failure was the government response, in general. A sense of underestimating the force and brutality of the storm was present. Only when the horrific images of people standing on roofs and crowding in anger outside of the Superdome did the government begin to do something which resembled effectiveness. The government report that came out at characterized the government's response as a "failure of initative." The overall problem to the government's repsonse was one where there was a "perfect storm" of misjudgement and inefficiency.