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Personally I think bronze does taste nicer, but that could just be where we buy it from rather than the type
White I think came about because of the fashion for perfect looking white meat, a bronze turkey will always have dark marks from the black pin feathers. They also tend to be slower growing breeds, so less quick to turn a profit (plus they'll eat more in that time so cost more to produce), and not the OTT massive breasted shape you tend to see in adverts showing the "ideal" Christmas.
Bronze has had a resurgence in recent years, no doubt helped by the celeb chefs, and the fashion for "heritage" stuff, and therefore the associated increased price tag!
I have had turkeys from all sorts of places and can say honestly that we have never noticed much difference. Consequently I just go to the supermarket and buy one, saves a fortune compared to the butcher.