In the first case, Rumsfeld compares Secretary of State Colin Powell to his wife.  Although Powell is put in a feminized position by this comparison, Rumsfeld's masculinity would also seem to be threatened with the egalitarian presentation of his wife as an oppositional figure.  Given "Western" assumptions about male chauvinism in the Arab world, this is also an interesting statement in that it puts Rumsfeld in the position of the henpecked husband before an audience that allegedly takes a vertical relation between man and wife as the norm.
 

I work with Colin Powell every day. We have views that are very similar on most things. We differ from time to time, but then I differ from time to time with my wife on various issues, so that doesn't mean much.

In the second case, Rumsfeld contrasts the press's opinion of him, as a "bad guy" in the situation with Afghanistan and the Middle East with the less biased opinion of his family.  He defends himself as a "nice person" and suggests that his wife and children would testify to the goodness of his character.
 

I don't know, there's something about the press that they like to get up in the morning and create conflict between people. It's apparently a lot easier for people in the media to write about personalities than it is about concepts and strategies and direction.  If you personalize a thing into good guys and bad guys, it's an easier story, I suppose, for a journalist, but it's not terribly useful.  I've been kind of amused by it all from time to time, and my wife and children know I'm basically a nice person.

In both cases, Rumsfeld introduces his wife as only a figure of potential rhetorical opposition, but one whose explicit opinions and own words do not enter into the public sphere.