Most people can't tell the difference between a bleached and an unbleached flour. If you can then you should obviously choose the one that tastes best to you.
The only place it makes a huge difference in performance is when a flour is bleached via chlorination. You get much MUCH lighter cakes with a bleached cake flour than with unbleached. You will get soup instead of bread dough with a chlorinated flour though, LOL!
There is a heat process that approximates the effect of chlorination on flour, you can google "Kate Flour" if you want to read about a home method approximating this process. Unfortunately no flour is so treated in this country, to my knowledge (this country being the USA), and even in the EU where it is being produced it is not available to home bakers but only to commercial bakeries in commercial quantities. Since chlorination of flour is one of the many maturing processes that is banned in the EU, cake baking has become an exercise in disappointment for many who yearn for the light fluffy cakes possible before the ban. Hence, the development of "Kate Flour".
There is no difference in performance between flours that have been merely whitened (eg where the bleaching agent has no maturing effect on the flour) and those that are "unbleached" - when the flours being compared are the same in all other respects. However, many bleached flours in the USA are blends of soft/white spring/summer wheat varieties, while many (but by no means all) unbleached flours are intended for bread and are usually comprised solely of hard wheat varieties with the higher protein content that make for better bread.
Pillsbury's bread flour is 100% hard red spring wheat and is nominally 12% protein; I'm not sure how tight the tolerance is on that, I couldn't get them to tell me. I would guess it varies around 12% by probably +/- 0.25%, at least.
GM bread flour is a mixture of hard/soft winter wheat and is nominally in the range of 11.7% to 12.3%. Again, I do not have any information on how tight the tolerance is but it varies by at LEAST that much. Also, the varying amounts of hard vs soft winter wheats introduces additional variation in performance due to a fluctuation in the proportion of gliadin vs glutenin. It's perfectly usable for bread, don't get me wrong, but it's a bit temperamental for something like artisan loaves or (especially) a baguette.
I think the GM flour is bleached but the Pillsbury bread flour is not. I'm not positive though; I didn't write that down in my notes.
King Arthur bread flour is 100% hard red winter wheat. It is unbleached. It is 11.7% protein, within 0.02% tolerance (very tight, and hence very uniform between batches).
Although hard red spring wheat is nominally a little higher in protein content than the hard red winter wheat, many bakers prefer the winter wheat nevertheless when baking bread because most varieties have a slightly different proportion of glutenin to gliadin (the proteins mainly responsible for the development of gluten networks). The proportion of gliadin (helps with extensibility/stretchiness) to glutenin (helps with elasticity/recovery) is felt to be more favorable in general in most hard red spring wheats; hence the thought is you get better oven spring and a better handling dough.
Which brings us back to relative performance in bread. Many of the "tests" of bleached vs unbleached flours that I have seen reported are comparing King Arthur AP flour to GM or Pillsbury AP flour (or both) and there just is no comparison because they are totally different types of wheat. They report the difference in performance is because the KAF AP flour is unbleached and the other two are bleached, but they are actually not even comparing apples to oranges, they're comparing apples to pomegranates. KAF AP flour is not REALLY an "all purpose" flour, it's a bread flour.
Mainstream retail AP flour produced for consumer sales runs right around 10% to 10.5%, with a range of 9.8% to 12% - you will generally find it on the higher end of the scale in the northern USA and towards the lower end of the scale in the southern USA. What this means is that sometimes - sometimes, but you can't be sure when - you can get a decent bread out of a retail AP flour, but sometimes all it's good for is biscuits. Note that regional northern brands such as Robin Hood or Dakota Maid are more similar to KAF AP flour, but the national brands vary too much to turn out reliable bread.
More appropriate comparisons of KAF AP flour to actual bread flours are less clear regarding relative "superiority" for bread. Generally KAF AP flour is felt to be superior, but that result is moderated by the fact that KAF AP flour is a much more uniform and reliable product because of the much tighter standards at the mill. That's a real effect but it's not due to bleached vs unbleached.
When I can't get KAF AP flour, I prefer either Con-Agra bread flour from Costco or Sam's, or failing that, Pillsbury bread flour or a mixture of Pillsbury bread flour in a 3:1 ratio with the GM bread flour. I don't care for GM bread flour in general. I would like to try the White Lily bread flour one of these days, before we move out of the area to somewhere where I can't get it any more. There are undoubtedly better (or at least cheaper) flours for bread but they're only available commercially and I don't have a source.
Anyway. Basically, for those bleaching processes that have no additional maturing effect (the only commonly used such agent among retail level consumer flours currently being chlorination), there is little or no measurable effect when comparing the same types of flour.