I am putting up some wainscoating and a bench in our mud room that I
intend to construct out of white oak (veneer plywood for the
wainscoating and solid for the bench). The finish will be light stain
and poly for protection.

White oak plywood is available in both "Quarter Sawn" and "Flat Sawn"
with the Quarter Sawn costing 2x as much. I am willing to pay the
extra money if the result is worthwhile but am not sure how noticeable
this is in paneling having never used Quarter Sawn. Any pointers on
the right way to do this?

Thanks