For example I just simulated some bad data. A has 480 observations and a mean conversion of 33%, B has 410 observations and has a mean conversion of 37%. The p-value here is 0.0323 In the traditional A/B testing model we'd be done and claiming better than a 10% improvement!
However when I sample from these 2 beta distributions I see that my credible region is -2% to 34% meaning this new test could be anywhere from 2% worse to 34% better. No magic value is needed to tell you that you really don't know anything yet.
Another huge help is the use of a prior. Until your data overrides your prior belief you aren't going to see anything. Going with the last example, if I had a good prior that the true conversion rate on that page was actually 33% I wouldn't have even gotten a p-value of less then 0.05. On the other hand if I had a strong prior that the conversion rate was 50% that would imply that both A and B were getting strangely unlucky results, which would actually boost the probability that B was in fact an improvement.
On the philosophical side, Bayesian statistics are simply trying to quantify what you know, not give you 'yes'/'no' answers. Maybe the gamble of -2 to 34 is good for you, or maybe you really want to know tighter bounds on your improvement and aren't comfortable with any possibility of decline. Bayesian statistics gives you a direct way to trade off certainty with time.