The budget scale is pretty much meaningless these days. There used to be a time when more money spent (in regards to game-development) either meant, more time for development, more people working on a game, or both, (usually) resulting in a better quality game. That's why people started categorizing games into A, AA, AAA. For example, Final Fantasy VII was definitely a AAA game back then, with a staff of 100 to 150 people working on it, at a time when the usual dev-team sizes were like 20 people.
The high budget helped to make the game truly marvelous in many aspects: size, visuals, presentation, music, story, gameplay. It also directly translated into a much shorter development time, since it only took FF7 around one (just "one") year to be finished. This wouldn't have been possible with a smaller team - at least not this quickly. However, at the end of the day, it was the quality of the team and their directors, writers, producers etc. that actually made the game so great (money well spent I'd say).
In the last 15 years or so, AAA budgets have been much more common as well as huge development teams (100 to 300 or more people), but it didn't necessarily mean the games got better. In fact game quality has been all over the place and development time got much much worse.
I would even argue that "AAA" has lost much of its positive connotation today, as it mostly stands for: severely restricted creative freedom in development, very long development times (focus on too much presentation and graphical set-pieces to make it look more "premium), thus (often) resulting in overly formulaic, generic-looking games with expensive presentations that (often) end up not being that good.