I found Sega Genesis / Mega Drive 101: A begginners guide on racketboy.com a great resource. The name is just a trademark issue, it was the Megadrive, but was marketed as Genesis in North America since the name had already been registered.
The differences between the three models are basically:
Model 1 - Only one to include volume control slide switch for the stereo sound output. Suposedly, the easiest to perform modifications on it for regional bypasses, overclocking, LED change and S-Video output.
Model 2 - Smaller, simplified design. Stereo output included in the A/V outputs andno volume control for the sound.
Model 3 - Even smaller, harder to hack, not compatible with Sega CD or 32X.
There's even a forum thread on which of the three is the best model.
Regarding cartridge compatibility, the games are per region, but there's a good point on this:
System modifications such as adding
region switches and SCART outputs can
be readily accomplished on most model
1 and model 2 hardware.
Coincidentally I was considering buying a Model 2, which is also the one with the most votes on the mentioned forum thread, so I guess this pretty much answers my question.
UPDATE: There's even more information on the different Mega Drive systems on Wikipedia:
Variations of the Sega Mega Drive:
During its lifespan, the Sega Mega
Drive quite possibly received more
officially licensed variations than
any other console. While only one
major design revision of the console
was created during its lifespan, each
region has its own peculiarities and
unique items, while other variations
were exercises in reducing costs (such
as the removal of the little-used
9-pin EXT. port) or expanding the
capabilities of the Mega Drive.
The article covers all of these variations.