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They are really not the same thing or concept!

 

The double wheel and co-axials all come from A.L. Breguet's Natural Escapement, a lubrication-free escapement with clear advantages over the swiss lever. Unfortunately, there was an issue of backlash in the gears that Breguet couldn't solve with the engineering of his day and he had to abandon the movement.
You may want to dive deeply into it, but the short answer is that KV's version has the same issues as the original, but with today's fine tolerances, these are small enough that you have an accurate wristwatch. KV's daughter is learning the trade, so maybe she will take over!
George Daniels solved the problem entirely. First with his genius double impulse impulse chronometer escapement (used in his famous Space Traveller, which you should look up right now). With the engineering of his time, he thought it was too large a movement for a wristwatch. The 42mm, 10mm thick Frodsham is the first ever wristwatch to use this movement. It is truly lubricant free and should have excellent rate stability.
The brillant co-axial was George Daniels' way to take his ideas and apply them in a form that could more easily fit a wristwatch. Wristwatches with this escapement tend to be on the thicker side, and require a tiny bit of lubricant in the escapement if silicon parts are not used. However, they are known for having excellent rate stability.
That's the short answer, I'll let more knowledgeable folks chime in with additions/corrections.

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