During President Obama's speech, while he talks about works like "we", "binding", "strength", "legend", he raises his hands (sometimes only one hand, sometimes two hands together, which I think depends on the scale of the emphasis he tries to put on his statements) a little bit, and then pushes it/them down with some force. I think he does this both to make his speech more lively (not just standing there with no body movement) and to put emphasis on his statements. His facial expression is serious, which is effective to signal audiences to pay attention to his speech and also to demonstrate that his speech contents are serious. His eyes turn around and look at different directions while he is talking, again to make his speech more lively and to cause audience's attention by making eye contact with them.
Comparing the two speeches, Rice has more head movement in her speech, and she looks down to her speech paper much more often than Obama. Also Rice's facial expression is less serious than Obama