I agree with this assessment by Noah Smith: it is not powerful at all. It can seem powerful at times, like a very high wind. But like a very high wind, it either subsides or moves on. Sometimes there is damage, but mainly not.
If you disagree, I recommend you read his piece. It’s a pretty strong argument for why twitter as a social force is limited.
P.S. I have felt that for some time. I mainly post things that are either positive or amusing. If I want to take social action, there are concrete ways to do that.
P.S.S. Tweets are like straw, blowing this way and that way, yet not moving and not affecting things, besides making a nice noise.