It's easy to prove Borel-Wadge determinacy from Borel determinacy. But it's often said that Borel-Wadge determinacy is 'much weaker' than the latter. This is then argued by showing models in which the former is true but the latter false.
But is there also a proof of Borel-Wadge determinacy that doesn't use Borel determinacy, and in that way shows it's 'much weaker'?
Yes (but I did not check the proof myself). Unlike Borel determinacy, Borel Wadge determinacy is provable in second order arithmetic:
Louveau, A., and J. Saint Raymond. “Borel Classes and Closed Games: Wadge-Type and Hurewicz-Type Results.” Transactions of the American Mathematical Society, vol. 304, no. 2, 1987, pp. 431–467.