There's a pretty complex answer to the question, but I'll try to give some precisions.

All are names of regions, but they changed so much with time that it's a pretty hard thing to be succinct when talking about them.

Simply put, Aquitania is the land of the Aquitains people, the most common of them being the Auscii, in Antiquity.

The name Guyenne is a deformation of the previous name, which became Guiana in Occitan, or Guyenne in French.

Lastly, Gasconha, Gascogne or Gascony is a name that comes from the Gens vasconum, which is associated with Basque culture. The W sound attached to the letter V here, in French Occitan and other Latin languages, changes to a hard G : William/Guillaume, for example. And so, in time, we have Gasconha, the land of the Gascons.

Politically, only Guyenne completely fell out of favor. And I'll skip it for today.

Aquitaine was the name of a Kingdom and the renowned and powerful Duchy, and is still the name of a French region, though it has recently - but only a little - changed.

Gascony was also a name of a Duchy, but its reputation is due to many things.

It regained a lot of popularity lately.

In part caused by some of the heroes of France, like La Hire, or the Maréchal Lannes.

Also, in literature, the name, for many authors, also was a romantic symbol for the stereotypical Southern Frenchman, as exemplified by the use of D'Artagnan and Cyrano de Bergerac, both of whom existed, but weren't as you'd suspect, the same as the literary heroes. For example, the real Cyrano wasn't even a Gascon.

Finally, Gasconha is still relevant as it is the place where Gascon is spoken. Gascon, a brother or a cousin of Occitan, depending on the expert, is still a fascinating language, spoken in France and in the Val d'Aran.

All in all, it boils down to terms that had their definitions change with time, as the region they were supposed to represent changed as well. Those were the facts.

On a more personal note - none of what follows is objective -, however, I would propose that the relative popularity of Gascogne has to do with its role and connections to the English-speaking world. And more precisely, with English tourism.

The Three Musketeers are still very popular in said world. And Gascogne, or more precisely the South West, has ties with England since it gave them some of their more known rulers, i.e. Richard Lionheart (even though he spoke mainly Limousin, not Gascon), and his mother, Aliénor. Lastly, Gascony was an ally of the English during the Hundred Years War.

This created the perfect recipe for Anglo-Saxon to hear about the region, even though the name, Gascony, wasn't much in use anymore within the French community. Thus one may think that the English made it popular since they are the most avid visitors of the region, and are the main offender when asking about it by its name. The most evident example of this is maybe the fact that the Landes region changes its name to Landes of Gascony, even though - while it certainly has a culture derived from Gascon culture - the name never really was attached to it before, which speaks volume about the attractive value and connotation of the word Gascony, mostly for tourists.

Sorry for the wall of text. Hope that helps you !