From Old French Gascoigne, from Gasconia, from Wasconia, from Vasconia from Latin Vascones, plural of Vasco, from Ancient Greek ουασκωνους (ouaskōnous), as used by Strabo in his Geographica (23 CE), Book III. either from αἴξ (aiks, “goat”) (literally “he-goat people”), or variant of Ausci (from which Aquitan, Occitan, and perhaps related to Euskara). Cognate to Basque (compare to Gascon).
For sound changes: /v/ → /w/ occurred in the change from Latin to Proto-Romance, while /w/ → /g/ 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, under influence of Basque, hence the divergence.