The terms "reason" and "rationality" are generally used interchangeably, where the latter is perhaps more technical, or sometimes "reason" is used to describe the human faculty while "rationality" the normative standard to which the faculty aspires.
"Reasonable" has acquired a more general usage in social discourse as anyone willing to listen to reason, i.e. anyone whose motives you can figure out. The practical upshot is not unlike being "agreeable" or "amiable:" someone who is unreasonable is irascible and contrary, and so probably not someone you want to deal with if you can avoid it. Saying someone is "irrational," on the other hand, is plain old condescension; you're not just saying the person is stubborn, but to some extent stupid.
Rand claims to be rational, but not "reasonable" in the sense of amiable: an Objectivist, she says, will compromise about matters of no moral import (what movie to go see, for instance), but not even in the slightest with the dictates of her morality. So if there's public pressure to give to charity, which to her by definition is putting someone else's needs above your own, then bowing to that pressure, even a little bit, is a self-betrayal. (And it matters little to me whether she has a loophole whereby it's OK in some circumstances to give to charity; pick another example if you know enough about her views to do so.)
On the other hand, someone with an "open mind" will claim to be reasonable, to listen to all comers and truly try to internalize what sense they may bring to the table, but are open to the charge of irrationality. Emerson stated "A foolish consistency is the hobgoblin of little minds," which Rand criticizes him for (leaving out the word "foolish" in her paraphrase). I always took this to mean being open to growth, non-dogmatic in the spirit of my previous post. Accepting that one's self is something that is developed over time, there's still the question of whether this development happens primarily through adolescence and young adulthood, or whether it is sensible to talk of growth in the same sense as an already mature adult. Integrity, for Rand, i.e. having and maintaining an authentic self, is about holding true to a position, whereas I tend to view ideas as less central in shaping the self.
This might sound strange coming from a philosophy fan, and would require a lot of caveats to develop as a full-blown position that I'd assent to consistently, but think back to Lacan and Freud here. As Wes stated in our Rand discussion (a criticism also reflected in the quote from Rand's psychologist paramour Nathaniel Branden), you can't in a straightforward way "program your brain like a computer." The extent to which a philosophy really "sinks in," the things in it that underlyingly appeal to us that might make it sink in (or not), and how even inveterate beliefs actually influence behavior are knotty and indirect, and anyone (whether a Randian or Lacanian) who professes to have a formula to manipulate the works is invariably engaging in a kind of blundering alchemy. As bad as such professional voodoo doctors are, the amateur (i.e. you) is in an even worse position. This is not to dissuade people from trying to understand and control yourself, but know that you're doomed to be partially examined, which hopefully will be enough to avoid overt stupidity.
So of course we try to be rational, amiable when in doubt (which in practical situations there usually isn't called for), and usually forgiving even when not in doubt (Rand considers it a violation of personal integrity not to pass judgment on everything you see, but Peikoff at least recognizes that simple politeness means you don't have to voice such judgments in many situations). Our dissing of Rand in the episode doesn't mean that we reject rationality in the broad sense, which in most cases is behaviorally equivalent to reasonableness but which in philosophy is useful precisely to critique the everyday. As Nietzsche recognized, even though our Reason is pathetically limited compared to the Enlightenment picture of it, it's basically all we've got.