While the words are related and synonymous in many contexts, they don't quite mean the same thing in all situations.
For example, a slow cat can have quick reflexes.
One person on Yahoo Answers wrote:
I like quick cars but not necessarily fast.
I would take this to mean the writer enjoys a car that can move quickly out of an intersection, but doesn't necessarily cruise at a high speed on the autobahn.
Had the writer said, "I like fast cars but not necessarily quick," then I would assume they would want a car that could easily travel 200 kph on the expressway, but wouldn't care if it's a little sluggish getting started at a red light.
Also, some adjectives simply get associated with certain nouns more than others. And sometimes synonymous words get paired together for emphasis. In a sentence like the one you quoted, for example:
"Relief workers need to be quick and fast," he said.
The speaker is conveying a sense of urgency, and not necessarily expecting anyone to parse those two words carefully, and discern some subtle difference between two.