Not very much. The SNES was the king of stuffing coprocessor chips onto cartridges; the hardware was a minor 6502 upgrade paired with amazing sound from Sony, and some pretty good video hardware. Sega did almost the opposite: they took the 68000, a pretty good CPU for the time, and paired it with OK sound and a tweak to the TI VDP they were using in the Master System. The average SNES game just needed the extra oomph that a coprocessor could give, while the average Genesis game probably was fine with the 68K.

That's not to say coprocessors didn't exist. Sega loved expanding the Genesis - but with hardware-addons that didn't sell very well. The Sega CD is effectively a second console's worth of coprocessors. It included much-improved audio hardware (which barely got used because you had CDDA), another 68000 CPU, and some fancy rotozoom ASIC stuff. There is a SNES style "just sneak a chip onto the cartridge" thing, though, but there's only one game that does that: Virtua Racing. It has a very primitive hardware rasterizer on board, which actually causes problems with some later-model Genesis consoles. It also won't work at all if you have a 32X plugged in.

Stuffing more chips onto game cartridges was a thing that happened with the SNES quite a bit, but really only happened with Virtua Racing on the Genesis/Mega Drive. If you look at early titles for a particular platform vs the later ones, the later ones almost always look/perform better because the developers learned how to milk more resources out of the platform. An example from the Playstation 1 era is Naughty Dog Studios, as seen here in this old HN thread: https://news.ycombinator.com/item?id=9737156

The other replies describe the lack of coprocessors, but there is one way in which Mega Drive cartridges got more sophisticated over time - increased ROM capacity.

At launch, most games were around 512KB in size. By the end of the machine's lifespan 3MB was common and there were even some 4MB and 5MB games.