One significant, one not, so A beats B?

Two strategies are evaluated. Strategy A has an estimated edge of $2.5$ bp with standard error $1.0$ bp (so $z_A = 2.5$, $p \approx 0.012$, significant). Strategy B has an estimated edge of $1.8$ bp with standard error $1.2$ bp (so $z_B = 1.5$, $p \approx 0.13$, not significant). A manager concludes, "A is significant and B is not, so A is meaningfully better than B."

Test whether A's edge is significantly greater than B's, and explain the fallacy in the manager's reasoning.