Leonardo DiCaprio did not receive an Oscar nomination, despite being nominated for a Golden Globe, Sag and BAFTA, for portraying William "Billy" Costigan, Jr. in The Departed.
This is actually one lack of an Oscar nomination that is extremely easy to figure out since DiCaprio was nominated for his performance in Blood Diamond. It is rather odd to note, something I don't know if that many people remember, is that DiCaprio was actually double nominated at Sag. The problem was he was not double nominated in lead, but actually received a nomination in the supporting category which is about as ridiculous a case of category fraud as one can imagine.
Anyway DiCaprio takes on the role of the cop who goes undercover as a criminal that was originated by Tony Leung in Infernal Affairs. Of course one of the reasons The Departed works as a remake is it smartly takes many different approaches around the same plot. One of these is in the character of the cop which is written and played very differently by DiCaprio than it was by Leung. Leung's take on the character was more based around the exasperation of being an undercover cop, as he's already been one for 10 years by the time the plot begins. The Departed differs greatly here by starting at the beginning of his time on his assignment and going from there. DiCaprio's work therefore is one quite the different wavelength as we start with him in the film as he basically is told by his superiors that he'll never make it far in the police department with his background so he might as well settle for being an undercover cop.
DiCaprio is quite good in his earliest scenes in establishing Billy Costigan's always troubled personal background before he even gets into the deeper trouble of playing the criminal. DiCaprio carries himself with a quiet intensity to start reflecting the difficulty in Billy's life even before he was a cop. DiCaprio importantly in this scenes creates an emotional honesty about Billy before he must go into a world of constantly duplicity. Unlike his co-star I am about to get to, DiCaprio portrays Billy as someone frankly we as the audience can trust. Although obviously he will be hiding a lot as the film progresses, DiCaprio does not really hide Billy from us. He plays him as a genuine man whom is easy to sympathy from the scene where Billy is given the blunt news by his superiors. DiCaprio effectively makes the hero to go into the dark world of crime, which cannot be said about someone else who plays the criminal pretending to be the cop.
Matt Damon did not receive an Oscar nomination for portraying Staff Sgt. Colin Sullivan in The Departed.
Matt Damon's performance also greatly differs from his Hong Kong counterpoint, although in this case that is not necessarily a good thing. Andy Lau played his corrupt cop working for the crime boss as confidant and cocky but quite charming. Damon on the other hand plays him as confidant and cocky but not at all charming. Well that's not exactly quite right because I suppose Damon plays him as a guy who thinks he's quite charming but the problem is he really is not. It's hard to tell whether or not Damon made this an intentional part of the character. The only thing is he's just extremely unlikable here as Damon plays him as an especially contemptible sort. Damon pushes on the charm offensive like Sullivan really is always trying extra hard to be this flamboyant guy everyone loves, but there's just something so obviously despicable in his excrement eating grin.
Damon and the writing seem to purposefully paint Sullivan as a despicable villain who is there just to be hated. Damon seems to try every attempt to distance us from his own personal story, and simply look at him as the rat Sullivan is. There is a persuasive smugness in his depiction of Sullivan that makes you just simply want to see something bad happen to him sooner oppose to later. The thing is though Damon does not even quite make Sullivan the type of villain you love to hate, because he's just so cool while being just so evil at the same time. At the same time Damon also makes Sullivan seem as pathetic as he can as well as he seems to imply that's he's a spoiled brat more than anything. This is seen from his just look of pouting when clearly being impudent, or possibly even worse when he basically throws a hissy fit at Mark Wahlberg's Dignam when he won't give Sullivan what he needs to be able to find DiCaprio's character within the criminal organization.
Speaking of which DiCaprio makes use of the scenes, not in affairs, as Billy attempts to join Frank Costello (Jack Nicholson)'s criminal organization by connecting with his old neighborhood. DiCaprio is effective in these scenes by performing a fairly subtle act that he puts on as Billy becomes the criminal. DiCaprio does not accentuate it too much which is important since it should be believable, but he does carefully differentiate Billy's act from who Billy truly is. In the act DiCaprio is effective in creating the thug personality for Billy. That being the guy who stays pretty reserved most of the time, but is quick to fits of anger whenever he is even slightly insulted. DiCaprio is good in showing kinda the act as he has the moment where he purposefully kinda show Billy creating point of having a short fuse while technically not really being the thug that would connect with. DiCaprio is great at kinda making Billy seem like the not especially smart thug, while still conveying the true intelligence of Billy underneath.
DiCaprio's best moments though are portraying the absolute truth about Billy as he must do his job which is basically going on routine beatings and sometimes killings with Frank's thugs, while avoiding being found out which would cause his death immediately. DiCaprio portrays the growing stress of his predicament throughout the film incredibly well. Some of DiCaprio's best moments are his reactions as he sees the heinous acts depicted by Frank and the others. Honestly I would say DiCaprio even does his very best to make Jack Nicholson's over the top performance work by giving realistic reactions to his insanity. I won't say DiCaprio is able to completely make up for what Nicholson does, because Nicholson goes a bit too far with some of the mannerisms he throws in, but DiCaprio steals the scene with his down to earth performance. DiCaprio's fear and intensity in fact is worth of something far more intimidating than some random weird looks, and making rat faces that Nicholson does in the interrogation scene.
Now Damon is also decent enough in playing against Nicholson as well, and although I might sound otherwise, I don't think Damon's performance is bad. He is good at being a guy you want to see some very bad things happen to, but in this vein his performance does not exactly go anywhere either. It seems like the film is attempting to show that perhaps Sullivan does have some regrets about his corruption, as well as feels like he is becoming isolated, the problem is Damon does not follow suit at all. Damon just seems content in being the jerk, and there is never a hint of regret in his portrayal, which was the most remarkable quality in Andy Lau's version of the character. Of course it did not need to be exactly regret, it could have been something even less which would just be in creating a sense of Sullivan becoming alienated from all others, but Damon does not even try to convey this. He's content with just being sleazy scum which he's not ineffective at to be sure, but the role does have deal of potential he never bothers to try and find.
DiCaprio though gives a powerful portrayal of simply the weight of having such pressure on the man for so long. DiCaprio is fantastic in internalizing the devastation in Billy as even greater tragedy befalls. DiCaprio is creates such a poignant depiction of a hero. DiCaprio expresses the emotional turmoil wonderfully and even does manage to bring a bit of charm to the role. He does not overplay it, that would be all wrong for Billy, but he just adds another reason to sympathize with Billy's plight throughout the story. DiCaprio earns the tragic nature of the character, and makes his exit devastating as it should be. Now Damon, unlike Lau in the original, is not standing up in his final scene so to speak. The only thing that can really be said when Damon makes his exit is a loud "FINALLY" due to just how unlikable he makes Sullivan. The ending is satisfying because Sullivan simply gets his comeuppance, but perhaps there could have been more complex emotions elicited from that scene if Damon had bothered to bring more complexity to the role. Damon never matches DiCaprio, and unlike the first film where you could become invested in both characters this is far more black white. The film could have hit even greater heights if both characters where as compelling as DiCaprio's Billy Costigan, but that never happens as DiCaprio gives a great performance while Damon does not, although he technically does not give a bad one either.

(For Damon)
(For DiCaprio)