From Old French Gascoigne, from Gasconia, from from Vasconia from Latin Vascones, plural of Vasco, from Ancient Greek Οὐασκώνων (Ouaskṓnōn), as used by Strabo in his Geographica (23 CE), Book III. either from αἴξ (aíx, “goat”) (literally “he-goat people”), or variant of Ausci (perhaps related to Euskara). Cognate to Basque (compare to Gascon).
For sound changes: /w/ → /v/ occurred in the change from Latin to Proto-Romance, while /w/ → /g/ (in loan words starting with /w/) common in (non-Iberian) Romance languages, notably French; compare warranty and guarantee, William and Guillaume. By contrast, /v/ developed into /b/ and /β̞/ in Spanish and Gascon (Spanish Vasco, IPA(key): /ˈbasko/), hence the divergence.