The Genesis 2 is basically just a redesigned Genesis.
The Genesis 2 was much smaller and squarer then the original Genesis. It did not have the headphone jack, volume control or RF TV output. The A/V port changed to a custom multi out port that now provided stereo sound to T.V's (previously only mono was output by the Genesis, the headphone jack had to be used for stereo sound). The power lead port was made smaller and a different AC adapter was used. Finaly the power switch was changed to a push button.
Both the original Genesis and the Genesis 2 could be connected to the Sega CD and 32X add-ons. The Sega CD was even remodeled to better fit the Genesis 2. A specific version of the Power Base Converter was also designed.
The life of the Genesis 2 came to an end in 1996. Sega was focusing on improving sales of the Saturn and they were also working on the Dreamcast which was set to be released in 1998.
An even smaller Genesis 3 was released by Majesco in 1998.
Adapted from Miles Mason's Genesis 2/Megadrive II text