There advantages and disadvantages to both. Wood is actually slightly stiffer on the quarter-sawn plane. Bows made from quarter-sawn boards in the core can develop problems when making high string tension bows with low string angle at brace. The problem, and this is one I have noticed with Osage in particular, is lateral stability. With such bows you have to have string tracking pretty well perfect or else the bow wants to self-destruct, sideways.
In the early days of the RD BBO, bows were made out of Osage boards that had been cut from wood that was pretty well unsuitable for selfbows. This made for slats that had considerable ring runoff along their entire length. Not much of a problem once backed with Bamboo except that as you made your bow you would get quite a lot of "steps" from the early-wood being scraped down faster than the late-wood. I don't know if this affected performance, most likely didn't, but it sure was a pain for appearance especially if the early-wood ring was thick enough. You don't get this "problem" with quarter-sawn boards. The ideal would be to have perfect flat-sawn boards for the core but who wants to cut perfectly good Osage like that into boards?