What have been decisive weapons?

[The Wizard]

I'd say the bayonette was a bigger departure from the old ways then the machine-gun, see WW1 which was the old (bayonette) way of fighting a new war, didn't so much change tactics just the amount of deaths, while the bayonette instantly made the old ways absolete making the brittish unbeatable untill others adapted.

The bayonet did not give European armies the ability to conquer vast swaths of land with minimal forces. The machine gun did. The time when the machine gun and the repeating, bolt-action rifle were only available to European forces was probably the only time in history where tactics and strategy didn't matter. Once the opposing forces gained the technology, of course, that advantage rapidly disappeared (like in the Rif War, for instance).

Nor have I heard of the British being the first to use bayonets. As far as I know it was the French, and the idea was rapidly adopted by all major European powers in turn, making it widespread by the 1660s.

There had been projectile weapons on ships before that, and people had been experimenting in trying to fit as many and as big guns as they could on deck.

Not on the scope and scale that early modern Europeans did it, nor with the same reach and mobility, nor with the same firepower. European naval power allowed them to usurp world sea trade by the 17th century, long before they became economically dominant following the Industrial Revolution.

P.S. Army B would win. Army A cannot sustain a war effort on the same level that Army B can. Take the conquest of the Americas by the Spanish as example -- had disease not crippled indigenous society and had internal conflict not been properly exploited by the Spaniards, there would have been none.