1. Bayesian stats is an approach that tends to make model assumptions fairly explicity, whereas in frequentist approaches, many assumptions are fairly implicit (Normal distribution of data, etc.) 2. I would consider myself a Bayesianist but I am sceptical about too much mention of esoteric terminology like "Belief". Bayesian probabilities are probabilities following the Kolmogorov axioms, which is also the foundation of Frequentist stats.
For decades, Bayesian inference was impractical because we need to resort to sampling methods and (a) computational power was insuffient and (b) we didn't have algorithms like No U-Turn Sampler (NUTS).
Both aspects are 'solved', so why is Bayesianism not universally adopted? Of course it still has a reputational disadvantage, but I think more importantly its because
* frequentist methods are good enough for purposes of publishing research [] for some problems we really have a hard time assembling bayesian graphs * some inference methods (e..g. Kalman filter) can both be seen as frequentist or Bayesian
As a bayesianist I am amazed at how well frequentism can work, even when the 'traditional' way of applying it contradicts the derivations of founding fathers like Fisher/Pearson. It's almost as if we have an evolutionary process at play.
[*] That is, if you use p-Values as publication thresholds