Re: Eh? What?
Twitter is more than what is available on public feeds. I don't post a lot of public tweets and my profile has no information on it at all, but I do frequently direct message companies' support Twitter accounts (since that results in the fastest response). Direct Messages are supposed to be secured so they frequently contain things like account numbers, phone numbers, addresses, etc. A hacker can grab all that info very easily if they access your Twitter account.
Also, many sites allow "sign in with Twitter". Getting the Twitter user name and password could allow hackers to automatically log into Facebook, Yahoo, Google, etc.