To demur is to show reluctance or to hesitate, like not quite getting in the car when someone opens the door, but demure is always an adjective describing a modest, reserved, or shy person, and sounds like the mew of a tiny kitten.

Demur is usually a verb, but it can also act as a noun meaning "the act of hesitating or objecting, or indecision." It's like saying "no thanks" by ducking your head and walking away:

"There isn't any too much room in this bungalow as it is," demurred Mr. Pottle. (Richard Connell)

But strangely, when given an opportunity to be influential, many demur, backing away from the conversations that result in the making of decisions. (Computer World)

Not surprisingly, perhaps, Tett stepped out of her chair role's impartiality momentarily to demur at that. (Guardian)

Add an e to make demure, which describes a shy, polite, legs-crossed, hands-in-lap kinda gal:

Demure? I think you need to read the definition of that word, because no one in hot pink shorts and red manic panic hair is demure. (x17)

Canada, represented as a demure little maiden, stands just behind Hawaii, an interested spectator, apparently more than half inclined to follow his example. (Frederic Taber Cooper)

Demur when a cute trucker invites you to an all-night rave if you want to be considered demure. Ladies love an e. Demur sounds like dee-mur, while demure sounds like a baby kitten: de-myure. Mew.