I'll give my succinct spin: Throw a coin in the air 10 times. Say 6 times the coin lands heads up, and 4 times tails up.
A Bayesian would say: the most likely explanation is that the coin is biased in favor of heads with Prob(heads) = .6 and Prob(tails) = .4. Because given such a model, the chance of this outcome is 25% (10 choose 6 * .6^6 * .4^4). Less likely is the explanation that the coin is fair.
A Frequentist would begin by assuming a model, likely that the coin is fair. She would then argue that the probability of getting 6 heads in 10 tosses is about 21% (10 choose 6 * 1/2^10), and she would have tools (number of STDs from the mean, etc) to test how much of an outlier this observation was, in order to validate her model.
In summary, the Bayesian considers many models at once, and asks how likely are they given the observations. A Frequentist starts with one model, and asks how likely is the observation.
The Frequentist uses the scientific method: develop a hypothesis, then test it. The Bayesian does something else, which I think is better, but ill-defined. She somehow tests all hypotheses at once, and puts different confidences in each.