Here's how I see it. If you want to use diplomacy, you must have a credible carrot. If you want to use intimidate, you must have a credible stick.
Shut up, Hong. You know what I mean.
If you're facing demons, and you've got a great diplomacy rating, shouting, "Wait! I've got an offer!" might be enough to get them to pause in their attack if you roll really well. They'll ready actions to attack you if anyone attacks them, probably, and hear you out for one or more rounds.
If your offer is, "Let us pass, and I'll give you this shiny new bottlecap!" then no diplomacy check in the world is going to help you. If your offer is, "Let us pass, and I'll tell you the true name of your demon-lord's rival!" then you've got their attention. Talk with them, make another diplomacy roll. They get to make a diplomacy roll themselves to determine what constitutes a fair deal ("Nuh-uh. We let you pass, except that we get to eat the wizard's familiar, and you tell us the rival's name. --Okay, okay, shut up, Balthazar, plus we get to French-braid the dwarf's beard.") Then both sides can agree with the deal or not.
Same thing with intimidation, and here I can use an example from our game last night. I and my companions went charging into a room filled with a fighter, two ogres, and half a dozen humans. We dispatched a couple of their leaders handily -- but so many more soldiers were pouring into the room that we were soon outnumbered 8 to 1. An intimidate check here was damned near impossible, no matter how high my intimidate score was.
But three rounds later, we'd got the odds down to 2 to 1 against us, and when I, a tiny blood-soaked halfling, roared, "Run for your lives! A-ha-ha-ha!" my intimidate check was more successful.
All about the circumstance.
Daniel