Of course there would be differences between the populations. There would probably be genetically-linked psychological differences, too! But would B-islanders be more cooperative, because they needed to band together to survive? Would they be more competitive, because those who hoarded resources for themselves reproduced better? Would A-islanders be better at analytical philosophy, because ability to publish in analytical philosophy journals was sexually selected for? Would that have no impact, because analytical philosophy skill has no significant genetic correlate, or would they actually be worse at analytical philosophy because ability to publish in analytical philosophy journals is a poor proxy metric? Or would A-islanders decide that analytical philosophy nerds are undesirable mates, and select against such an ability? Would cultural factors on island B result in a better, safer society where a decade's worth of preserved food is kept at all times, and nobody has to venture out into storms, while the A-islanders become their own predators and destroy each others' food supplies until they suffer entirely-avoidable famines?

Would all these various effects cancel each other out, leaving no phenotypical variation except skin colour (due to the selective effects of melanoma and vitamin D deficiency), and no significant psychological differences except those caused by unselected genetic drift, which are utterly erased by the first generation of interbreeding? There is no way to tell, from the information you have provided. Your thought experiment reveals your (and my) own biases, nothing more. Thought experiments only work in mathematics and physics, not (except in extremely simple situations) evolutionary biology.

Agreed, there is no way to know the exact degree of differences between the two populations (although any geneticist would reject the notion of primarily unselected genetic drift in the example I described. I specifically outlined strong selective pressures in the environments). But at least we agree that they would not continue to be identical, and that the environmental pressure, over time, would also show up in genetic drift between the populations (there are surprisingly many blank-slaters who would disagree even with this premise).

We have tools in modern science to attempt to differentiate between different underlying components of genetic variation vs environmental variation (identical twin studies, adoption-at-birth studies, general population studies, longitudinal analyses, etc). In lieu of a 1000-year experiment (which we cannot recreate), we can make predictions. If factor X was primarily genetic, we would expect to observe A, B, C. If factor X was primarily environmental, we would expect to observe D, E, F. Then, we can basically incorporate various studies and results and do gradient descent, and observing which (means squared error) is smaller.

None of this is groundbreaking stuff, I know. But I have read arguments from both the primarily environmental camp and the primarily genetics camp, and one of them has to do a lot more mental gymnastics explain the differences between obvious predictions (under this paradigm) and the observed data in the real world.