With the Gameboy version you can raise up three tamagotchi at one time. You can change your Tamagotchi's Growth rate, save unfinished games so that you can return to them later and enter your Tamagotchi in tournaments. There are also more ways to care for your Tamagotchi than with the original toy! To start the game you can choose from three eggs to hatch, which will either revolve in a white or a black Babitchi. I especially like the black one, because it looks so funny when it poops (^__^).
The gameboy game has got more options than the original Tama, for instance you can feed it bread or ice and play three kinds of games with it. The growchart is similar to the handheld tama though. The music get iritating after, well lets say one minute ;-), so I'd suggest to turn it off. The good thing is that you can save your game and restart it whenever you want to.

You can still buy the Gameboy Tamagotchi on Ebay for about $6,50. You can also try to find the rom on the internet and try it out (for 24 hours) with a Game boy Simulator like VisualBoyAdvance.
Online Games
At FelixW you can play the Tamagotchi game online, use your mouse to raise it.

The official Bandai website gives you a change to play a cute Devilgotchi game, try to guess behind which door Debiru Mametchi is hiding. You can also play other games there, involving Osuchi/Mesuchi and Santaclautchi.