The foundations for the answer lie in the way you framed the question--the concept that Burgundy and the French could be different parties practicing diplomacy (and/or war) with each other. Because even as the fourteenth century saw the French monarchy consolidate and expand its power beyond the Ile-de-France, the dukes of Burgunday and their principality were coming into their own as simultaneously a subordinate of the king and as ambitious rulers in their own right.

Through the 1410s, Burgundy sort of tried to play it both ways. On one hand, they sought to make themselves a powerful voice at the French royal court. This protected the principality, but also gave the duke a degree of authority over France and over the monarchy's control (especially taxation) of Burgundy. On the other, they had good reason to maintain decent relations with England. "Burgundy" was actually two geographically distinct territories: Burgundy, which was sandwiched between France and the HRE about halfway down; and a second territory in the Low Countries, around Flanders. The Flanders-England wool trade, especially, was vital to economies.

The English victory at Agincourt and conquest of Normandy in its aftermath provided what Burgundy thought was its big chance. The French government in disarray thanks to the dauphin's flight from Paris (and, oh yeah, the duke of Burgundy murdering one of his most powerful supporters, the duke of Armagnac), Burgundy and a few remaining voices in Paris thought they could negotiate as France with England. However, the dauphin set up a regency from Bourges, and a delegation of his men murdered the duke of Burgundy right back.

England was now feeling a whole lot more "conqueror"-y (Burgundy, remember, was a key trading partner), and the 1420 Treaty of Troyes passed the French monarchy succession to England.

Philip the Good, the new duke of Burgundy, had every reason to be on England's side--and he was, solidly through the 1420s. But while the English alliance was fine for the Burgundian/Flemish economy, it wasn't doing much for Burgundy's political ambitions. The Parisian-French throne (as opposed to the Bourges-based one) passing from English butt to English butt excluded Burgundy from the role of power broker it had played in the earliest decades of the century, and the alliance with England and therefore English allies limited Burgundy's ambitions of geographic expansion.

These ambitions, for territorial gain in the Low Countries, ultimately sent Burgundy back to the France-based French side for the final decades of the Hundred Years' War. Manuevering to get Holland and Zeeland into the Burgundian fold through warfare, marriage, and brutal leverage was one of the major breaks with England. Jacqueline of Hainaut, the duchess of Holland and Zeeland (and Hainaut), was a Bavarian princess and in the orbit of the dukes of Gloucester; Philip more or less bullied her into making him her heir. This was a pretty clear sign that England (and its allies in the southern Empire) was no long the buddy-buddy of Burgundy.

The final seal was France guaranteeing Burgundy some of what it had long sought--power over part of the French heartland, specifically the region of Picardy just north of Paris.

From that point, Burgundy went on to spend a few decades positioning itself to become an independent kingdom, although that faltered and died along with Duke Charles the Bold in 1477.