As already noted, there are other brands that surpass Breguet measuring by certain criteria. Patek Philippe and even Rolex, for example, have a more continuous and independent history than Breguet. Other brands have higher resale value. Some companies (like JLC) manufacture more of their own movements.
But, to me, what attracts me to Breguet is the remarkable beauty of its watches. Unlike some others on these boards, I do not own any true "haute horlogerie" pieces -- no perpetual calendars or tourbillons or the like in my collection. The funds aren't there. In terms of simple timepieces, however, Breguets are wonderfully executed, achieving what I hope will be timeless beauty.
I currently own two Breguets. The first is a yellow gold 5140 -- the classique automatic with an enamel dial. The simple dial, with hand-painted numerals, is just beautiful. And the casework, including the characteristic Breguet ribbing, is equally wonderful. It is a watch, I think, that will never go out of style. My second Breguet is the ultra-thin 5157 in white gold. Again, what strikes me first about this watch is its dial; a remarkable guilloche dial, wonderfully executed. It is so simple, but on close inspection, remarkably sharp and detailed. And the 5157 also provides a view of the wonderfully finished Piguet-based caliber with a guilloche rotor. Really very nice.
There are plenty of other beautiful watches out there. Patek's Calatravas are wonderful; Langes are beautiful; Vacheron is making a line of beautiful dress watches; and there are many, many others. But of these, it is the Breguet style, combining wonderful casework, beautiful dials, and elegant hands that attracts me the most. Knowing that there is a nicely executed movement inside is also a plus (though the wonderful Piguet calibers in my two Breguets are not superior, to my knowledge, to the movements used in any of the other watches I have mentioned).
In other words, I just like them.