The TSE software will tweet questions and answers on (I assume) a periodic basis. This is fine for people who follow TSE on Twitter, but may not reach your followers.
You can tweet using the 'share' button.
When you tweet questions, people can go "Hey, I know the answer to that" and then become valued community members.
And when you tweet answers, people can go "Hey, I need to read that, I'll go over there and read it..." and this drives up the site stats which in turn gives a higher Google ranking.
All good. I can't think of a disadvantage to tweeting out your stuff (can you?) and there are lots of advantages.
Are there any aspects where tweeting your stuff results in a disadvantage?