... Republican and Democrats, is, in part, this:
The Honorable Richard Cheney was a United States Congressman, Secretary of Defense and the Vice President of the United States. Regardless of what you think of his opinions, or even think of the man personally, in formal circles, there is a certain dignity with which one treats the men who have held those offices.
On the Tonight Show and the Late Show, etc. etc., standup comedians do their work, and no one expects them to treat the subjects of their humor, often high elected officials, with any dignity at all.
One of the differences I would note between Republicans and Democrats is that I do not ever recall even Jimmy Carter, and some of his ridiculous statements, being treated with this sort of disdain by the Press Secretary of the last two Republican POTUS.
Perhaps I'm wrong, and would retract this if I'm shown so.
But to the larger point, it seems that Mr. Gibbs is incapable of distinguishing between entertainment and political leadership. Letterman is an entertainer. Gibbs is a (representative of a) political leader.
This inability to distinguish these would explain the Obama administration's preoccupation with Rush Limbaugh.