If you provide an online service – something that looks something up, or calculates something – you want to stop people from abusing/overloading the system. If the users need to proof they are human, automated scripts are at least slowed somewhat, so other people can get their turn.
For free online games you want to reduce the people who are able to create new accounts and play them automatically just to give themself help or farm random items.
CAPTCHAs aren’t terribly useful for stopping concentrated attacks, but they reduce the amount of spam that is sent via publicly accessible forms.
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