You have direct access to some incredible people. For a planet with 7 Billion people, this has democratised thought leaders. You can engage with Nobel Prize winners (@RobertJShiller and @TheDesmondTutu), celebrated scientists (@neiltyson or for some controversy @RichardDawkins, authors (@nntaleb and @GilbertLiz), politicians (@narendamodi) , comedians (@Trevornoah or @rustyrockets - Russell Brand), Philosophers (Nigel Warburton @philosophybites , @PeterSinger, @alaindebotton), Psychology gurus (@paulbloomatyale and @jonhaidt), Religious Leaders (The pope tweets himself @pontifex and the @DalaiLama) and celebrities of your choosing.
As confirmation that they really see your tweets... watch this very funny clip:
There is so much information on the web, it is daunting to find the stuff that is worth while. Twitter acts as the filter. You find people who are looking for stuff themselves. When they find something interesting they share it. If you find people who are genuinely sharing stuff they find interesting, this is different from traditional media which normally has a particular brand to present and an editorial bias. Here it is individuals - still biased but less constrained. You can use your friends and the world's thought leaders as your filter and you can help provide a filter for others.
As with real human interaction, I think the real power of twitter is when it is used as a conversation rather than a broadcast of just your own ideas. No one likes someone who just talks about themselves. Most people like people who are also interested in them. Most people don't like people who are unpleasant (don't be a troll). As a tool to participate in the important sharing of ideas without borders or thought police - twitter is the embodiment of free speech.