Original Genesis 1 systems (i.e. ones that were sold with Altered Beast) do not have the license lockout (formally called TMSS or Trade Mark Security System) and can play a handful of unlicensed games that newer systems cannot play (Budokan, Zany Golf, Onslaught, Ishido, Turrican, probably a couple others). It also has an extra port on the back that was never officially used outside Japan (and in Japan only for a couple very limited-release devices I think). As Borisz mentioned the audio/video connector is different, this is because the Genesis 2 supports a stereo A/V cable while Genesis 1 only supports a mono A/V cable (there is still stereo output at the headphone jack). Also the Genesis 2 uses fewer chips because a couple of the I/O chips were combined into one big chip.
In case you're wondering, the so-called "Genesis 3" is actually a clone not made by Sega (it is licensed by Sega, but not Sega's design), and it is missing some features that make it incompatible with a few things (from what I've heard, Gargoyles, Virtua Racing, 32X, Sega CD, most copiers, Power Base Converter / Master Mega Converter, and Game Genie do not work correctly with it).
