I would say that whether spring or fall is a better time to transplant would depend on whether you live in a warm or a cold climate, here in central Indiana, I can recall transplanting strawberry plants in early or mid April(this was a little bit earlier than what the people at the nursery recommended, so we had to provide protection against frosty or freezing temperatures),
So, what I can tell you is that in a cold climate like Indiana, spring would probably be a better time to transplant strawberries, and I would presume that in warmer climates that either spring or fall would work, hope I helped