Whereas Bayesians believe P(x) is odds at which they would gamble, so to speak, that a particular proposition x is true.
Is that it?
In contrast, Bayesians compute P(null hypothesis is false | prior knowledge).
Similarly, Frequentists compute confidence intervals (say at 5%), which is an interval that represents the set of null hypothesis you can't reject with a 5% p-value cutoff. In contrast, Bayesians compute credible intervals, which represent a region having a 95% probability of containing the true value.
Personally, I'm solidly in the Bayesian camp simply because I can actually understand it. To take an example, consider Bem's "Feeling the Future" paper  which suggests that psychic powers exist. From a Bayesian perspective, I understand exactly how to interpret this - my prior suggests psychic powers are unlikely, and my posterior after reading Bem's paper is only a little different from my prior. I don't know how to interpret his paper from a Frequentist perspective.
 http://www.dbem.ws/FeelingFuture.pdf For background, his statistical methods were fairly good, and more or less the standard of psychology research. If you reject his paper on methodological grounds, you need to reject almost everything.