Ask HN: What is the significance of the proposed ICANN transfer?

My mom recently asked me about the significance of the US handing over the oversight of ICANN to a yet-to-be-defined international body, and I didn't have a confident answer for her.

http://www.washingtonpost.com/business/technology/us-to-relinquish-remaining-control-over-the-internet/2014/03/14/0c7472d0-abb5-11e3-adbc-888c8010c799_story.html

The instigation for this move is apparently international distrust of the US gov't following Snowden. People have claimed that, through the control of the DNS root zone, an international ICANN could censor swastika imagery (e.g. France), political dissidents (e.g China), or dubious trademark infringement (e.g. US). Others have said that ICANN has essentially no power, and that all of this is a political distraction with minimal real-world effect. Still others have said that, by opening up the process, ICANN will become more transparent than when controlled by an imperfect US.

What are ICANN's actual abilities, and what are the possibilities for international control to lead to good or bad outcomes?