IMO it really didn't. At one point I thought of making a video showcasing neat tricks that Neo games do and I really couldn't find very many. Neo games mostly get by with brute force and a metric crap ton of graphic tiles.
I think the main advancement for Neo games came in the form of art, art direction, game design, and of course more ROM space. Things like the transition scenes in Last Blade, or the Satella News Network in KOF97 are really more about artists building those scenes than anything.
Rotation (like the Metal Slug's turret) is just done with more graphic tiles. Transparencies are done with sprite flickering and sadly that trick has not aged well (they did get better at it, Last Blade's flickering is more clever). Around 93 or so (I have the exact transition somewhere, but can't find it), almost all Neo games employed a double buffering trick by using two sprite pools. This allowed them to write to video ram more often, which probably allowed them to squeeze the system a bit more.
Puddle reflections (like the first stage of Metal Slug, or Italy stage in KOF94) are done with some clever sprite scaling. I really like this one, it's a neat little trick.
SVC Chaos and Blazing Star do some neat things with auto animations to pull off some impressive backgrounds (I made a video about this).
Super Sidekicks and Neo Turf Masters use the timer interrupt to pull off their perspective effect. But so does Riding Hero and just about every 2D racer ever made.
What else? I'd love to know of more. In the end what really allowed Neo games to be so impressive is how many graphic tiles they can use. Without bank switching and using a modern CHA board, the Neo Geo can work with 1.04 million sprite tiles. It's just a jaw dropping amount.
Its awful sound and limited colors aside, just giving the Genesis a healthy dose of graphic tiles enables it to come surprisingly close to the Neo. I think the RBS demo is a great example: