In the first scenario you gather some data and conduct the hypothesis test, the test tells you that the difference is so small that it didn't reach statistical significance.

In the second scenario, you start with some hypothesis (your priors), gather the data, and combine the two sources of information, to update your hypothesis. Bayesians don't use $p$-values, so they won't say that the result is "not significant", yet they can simply conclude that it is small. On another hand, Bayesian could define use region of practical equivalence and conclude that since the small difference falls into the region, it is "practically equivalent" to zero (call it "insignificant" if you wish).

There is no paradox in here. Both approaches would only differ in the conclusions if the Bayesian used strongly informative prior, that would influence the results, but if the priors are not "too" informative and the data is not too "inconclusive", this should not be the case.

In the first scenario you gather some data and conduct the hypothesis test, the test tells you that the difference is so small that it didn't reach statistical significance.

In the second scenario, you start with some hypothesis (your priors), gather the data, and combine the two sources of information, to update your hypothesis. Bayesians don't use $p$-values, so they won't say that the result is "not significant", yet they can simply conclude that it is small. On another hand, Bayesian could define some region of practical equivalence and conclude that since the small difference falls into the region, it is "practically equivalent" to zero (call it "insignificant" if you wish).

There is no paradox in here. Both approaches would only differ in the conclusions if the Bayesian used strongly informative prior, that would influence the results, but if the priors are not "too" informative and the data is not too "inconclusive", this should not be the case.