Bernheim is a straight wheat whiskey (>50% wheat, the remainder corn and barley.)
"Wheater" refers to a bourbon which uses wheat as the flavor grain (>50% corn, the remainder wheat and barley) as opposed to a more typical rye mash bourbon.
Bernheim doesn't have a lot in common with wheater bourbons. The wheaters you're likely to see around are the ubiquitous Maker's Mark, Weller line, and the Old Fitzgerald line (including Larceny.) If you're lucky maybe you'll also run into some Van Winkle bottles.