Well, the Netherlands by themselves wouldnt be Burgundy, terminologically. The name of the "Neo-Burgundian Empire" (so to say) derives from the Duchy of Burgundy, and the Netherlandic territories were only associated with it by dynasty. Of course, names can "wander", just look at Saxony. The Netherlands in any case were surely economicalyl way stronger than Duchy and Free County of Burgundy, so eventually I guess the Burgundian centre wouldve been there. Holding on to both could be difficult for purely geographical reason: if you want to connect Netherlands and Duchy/Free County then that corridor is an uneasy sandwhich between France and Germany... so it depends. I guess with enough luck it can be done, but its difficult to hold on to both for several generations. Of course, I guess, OTOH Neo-Burgundy might just expand enough that that corridor becomes broader than just a corridor...
Now, as for Burgundies, we have:
1) The original Kingdom of the Burgundians, a Germanic Tribe during the Great Migration Period who settled in the Rhone Valley. Conquered relatively quickly (that is, before the Alemannes and way before Saxons etc) by the Franks.
2) When the Frankish Empire broke down, the dynastic split lines went all through the territory of that former Kingdom. Eventually, two Kingdoms formed: The Kingdom of Upper Burgundy, and the Kingdom of Lower Burgundy, also called Arelate. Additionally, part of the former Burgundian territory stayed with the West Frankish Empire (i.e., France), and that was the Duchy of Burgundy.
3) Later the two Kingdoms united to the Kingdom of Burgundy, also called Arelate. However, it never was truly stable, with much strife for succession, and so crown eventually fell to to the (Holy) Roman Emperor, King of Germany and Italy. Thus, this Burgundy became one of the three kingdoms of the HRE, at least officially. De facto, few bothered to remember that in fact it was an own kingdom in the HRE - only very, very few HR Emperors had an own Burgundian coronation, for example. Thus, inside the HRE, it kinda dissolved, and most (and in the end, that is by modern day, err, all) was eventually gobbled up by France.
4) Thus the irony is that the one part of very-ancient-Burgundy to retain the name "burgundy" - was the part outside the main kingdom

That is the Duchy of Burgundy inside France. Thats the base of "Neo-Burgundy", led by a sideline of the royal House of France.
5) Well, okay, there was another part to retain that name, the Free County. Basically, some territories in Upper Burgundy fell to the Emperor personally, so they were made a seperate territory, and "Burgundy" was the logical name. Later, the territory fell to the Duchy (in personal union, that is).
6) What I in this post called the Neo-Burgundian Realm, consisting of Duchy, Franche-Comte and the Netherlands, basically. Basically, the dynastical expansion of the DUkes of Burgundy.