Very, very few debates, on the internet or off, result in either party being convinced of anything. If you feel strongly enough about something to publically debate it, you're not going to have your mind changed that easily. It does happen, but only rarely.
Thus, if you realize this, the real reasons for a debate can be:
- Convince onlookers, who are not as set in their views as either party in the debate.
- Sharpen your own arguments and wits - If you're going to hold an opinion, it should hold up to attack from someone who disagrees with you. Only discussing issues with those who agree with you makes you weak and unable to defend your viewpoints against those who actually oppose them. Your opponent will weed out your own weak arguments and fallacies for you.
- Related to 2, debating a subject will make your own thoughts clearer to yourself, and you will gain greater understanding of your own opinions.
- Arguing for fun. Pick a standpoint that you don't actually care all that much about, and get in an argument with someone else (who either does or does not care about the opposing view). This is a good mental workout, and will teach you many things about arguing matters you do care about. It will also teach you not to be too emotionally attached to your own views, which will only play against you in a real debate.
The internet has very little to do with it, except for providing ample opportunity for debate. The internet also has a more aggressive tone, due partly to people not being physically present and thus not holding back as much out of politness or fear. This is merely a small difference in culture, and doesn't affect the larger issues involved in a good debate.