AIUI, in begging the question a questionable proposition is slipped under the radar by putting it in the premise of some other argument. In circular reasoning, the silliness is taken a step farther, by using the conclusion of an argument to support its own buried premise.
T-Rex seems to disagree, though; the way he defines it, it's only begging the question if the thing being proven is itself in the premises, whereas I use Wikipedia's broader definition of "when a proposition which requires proof is assumed without proof."
Assume as an axiom that being always friggin' awesome implies being a pretty sweet dude, and also that the stated premises of the arguments are true:
Example 1: Premise: No one ever questions T-Rex's perpetual friggin' awesomeness. Conclusion: Therefore, we know that he's a pretty sweet dude.
Problem: Begging the question. There is an unspoken premise that T-Rex is always friggin' awesome. If the argument stated outright that "because nobody questions T-Rex's perpetual friggin' awesomeness, therefore he's always friggin' awesome", the flaw would be obvious. So it leaves this premise unsaid.
Example 2: Premise: No one ever questions T-Rex's perpetual friggin' awesomeness, and he's also a pretty sweet dude. Conclusion: T-Rex is always friggin' awesome.
Problem: Circular reasoning. The argument requires that being a pretty sweet dude implies being always friggin' awesome, which isn't the case. It tries to hide this problem by begging its own conclusion.
Note that even if it were the case that being a pretty sweet dude implies being always friggin' awesome, the argument would still be kind of broken: the unquestioned friggin' awesomeness is unnecessary. Unneeded premises are a sign of an argument that's not well thought out.
Sun Tzu said, "The supreme art of war is to subdue the enemy without fighting." This is also true in rhetoric. The best way to get a belief accepted is to fool people into thinking that they have already accepted it.
(Note, first-year students, that I did not say, "The best way to convince people of a belief". Do not try to convince people! It will not work; and it may start them thinking.)
An excellent way of doing this is to embed your desired conclusion as a presupposition to an enticing argument. If you are debating abortion, and you wish people to believe that human and non-human life are qualitatively different, begin by saying, "We all agree that killing humans is immoral. So when does human life begin?" People will be so eager to jump into the debate about whether a life becomes "human" at conception, the second trimester, or at birth (I myself favor "on moving out of the house"), they won't notice that they agreed to the embedded presupposition that the problem should be phrased as a binary category membership problem, rather than as one of tradeoffs or utility calculations.
Consider the recent furor over whether WikiLeaks leader Julian Assange is a journalist, or can be prosecuted for espionage. I don't know who initially asked this question. The earliest posing of the question that I can find that relates it to the First Amendment is this piece from Fox News on Dec. 8; but Marc Thiessen's column in the Washington Post of Aug. 3 has similar implications. Note that this question presupposes that First Amendment protection applies only to journalists! There is no legal precedent for this that I'm aware of; yet if people spend enough time debating whether Julian Assange is a journalist, they will have unknowingly convinced themselves that ordinary citizens have no First Amendment rights. (We can only hope that this was an artful stroke made from the shadows by some great master of the Dark Arts, and not a mere snowballing of an ignorant question.)