Character Analysis
Sandor
Sandor Clegane is a warrior who is sworn to protect and serve Joffrey. He does so, but he also starts comforting Sansa when Joffrey turns out to be a total jerk.
Scarred for Life
Sandor is horribly scarred, with the whole left side of his face "a ruin" from a little accident with a fire (30 Sansa 2.54-55: yep, two whole paragraphs about how terribly scarred he is). That little accident is named "Gregor Clegane," as Sandor tells Sansa (30 Sansa 2.56).
From that experience, Sandor probably takes the lesson that violence is the only real power in the world. And since it was his brother Gregor who scarred him and his father who covered for Gregor, Sandor probably doesn't care too much about family. This combination leads him to spend his days with someone else's family, being violent for them. How violent? He kills an unarmed butcher's boy without any apparent worries or second thoughts. Imagine what he would do if he caught Nymeria the wolf.
What's the Melting Point of a Golden Heart?
Although Sandor seems pretty scary, we see another side of him when he tells Sansa his family story. (True, he does threaten to kill her if she tells anyone, but he did open up enough to tell her the story in the first place [30 Sansa 2.65].) Also, when Gregor goes crazy and attempts to kill Ser Loras and his brother, Sandor shows some admirable restraint, at least in comparison to Gregor: "Thrice Ned saw Ser Gregor aim savage blows at the hound's-head helmet, yet not once did Sandor send a cut at his brother's unprotected face" (31 Eddard 7.82). So Gregor tries to kill Sandor, but Sandor never reciprocates – even after Gregor has ruined Sandor's face.
Sandor may be a violent and scary guy; and he may be a servant for monsters (Cersei and Joffrey: quick, which one is worse?); but he seems to have a heart of gold underneath that hard armor.
Gregor
Gregor is a knight sworn to serve Tywin Lannister. He's also a giant of a man, so watch out. Gregor goes berserk after losing a joust at the Hand's tournament and he leads the informal raids on Tully lands until the war breaks out officially. According to the rumors, he was also one of the Lannister soldiers who helped kill the last Targaryens, including the children (31 Eddard 7.71).
One Dimensional Gregor (Because One Is All He Needs to Kill You)
Sandor basically has three qualities: (1) scarred face; (2) violent; and (3) secretly maybe not a monster. By contrast, Gregor really has two qualities: (1) giant; and (2) violent. There's really no doubt that Gregor is a monster. For instance, Sandor tells a story of how Gregor burned off half of Sandor's face for playing with his toy – a toy that Gregor didn't even like. And when Gregor loses at the Hand's tournament, he kills his horse (in a really distressing scene) and tries to kill Ser Loras and Sandor, too.
Knights Are the Coldest
Plot-wise, Gregor is useful: he's a very dangerous person who works for the Lannisters. But he's also important thematically: he demonstrates a very clear lack of principles. That is, he swore all the knightly oaths about protecting women and the helpless, but he doesn't buy any of that. Gregor demonstrates to us that this is a world where being a knight doesn't mean you're heroic; far from it, in fact.
Very few knights take their oaths seriously. Barristan Selmy is one notable exception, but everyone else – like Jaime Lannister or Meryn Trant – is pretty much on Gregor's level. For them, might makes right. It's one of the main reasons that Sandor doesn't want to be a knight: if his brother the monster can be a knight, then the title doesn't really mean anything.
Gregor reminds us that we're not reading a romantic, happy story of an idealized medieval time. We're reading about violent people living in a violent time. In fact, it might be easier to be Gregor (without any conscience) than to be Sandor (with some secretly potential good inside).