Before I get started, I want to make it clear that I generally prefer the 2003 series to the 2012 series. Which is not to say that I hate the 2012 series, as I find that it has a lot of genuinely funny moments (and I actually love the art style and hybrid of Western cartoon/anime conventions in the show), but I believe the 2003 series has overall better writing.
"Things Change" (2003) & "Rise of the Turtles" (2012): The Turtles
The goal of every first episode is to introduce the setting, the story, and most importantly, the protagonist(s). So the question is, how well do the first episodes of TMNT introduce the titular Turtles?
"Rise of the Turtles" (RotT) starts off with a sparring session among the four. I thought it was a strange way to begin, because there is no context for the sparring session. Also, the episode clearly establishes a hierarchy among the four in terms of combat ability (that is later reinforced in the episode "Attack of the Mousers"): Leo and Raph are better than Mikey and Donnie. Donnie in particular gets beaten by Raph in a rather pathetic way. But it's problematic to introduce the Turtles hierarchically like this, especially when much of the tradition of the franchise is to emphasize that each of them is skilled in a different way. Instead, the first episode predisposes the audience (especially those who are new to TMNT) to favor Raph and Leo over Donnie and Mikey.
What do we get in terms of personalities? The first few minutes create the impression that Mikey is a bit silly, hyperactive and overconfident, without the skills to back it up; Donnie (again) is pathetic; Raph is the best fighter but also a little arrogant, and Leo is somewhat stoic and serious.
Contrast this to the beginning of "Things Change." First off, the 2003 series uses a short, voice-narrated teaser at the beginning of each episode, which I particularly love because it does a great job of previewing the episode with some action and hooking the viewer's attention. When the episode starts, Splinter is giving the Turtles a test, which only Leo manages successfully. This establishes him as the most skilled turtle, setting up for his position as leader. Note, though, that this has nothing to do with combat ability, but focuses more on Leo's ability to think strategically; the other turtles are no worse off for having failed.
The first few minutes of the episode does a lot to reveal the personalities of the Turtles and also their dynamics as a group. Raph is established as impulsive, straightforward, and confrontational; Mikey is somewhat laid-back, silly, and not as serious as the others about ninja training; and Donnie is the most knowledgeable one. The Mouser attack also shows off the Turtles' combat skills with more of a context for why they're fighting, as well as generally more action in the episode. In the aftermath of the attack, Leo is shown as the leader by default when he takes charge of the situation, while Don again is the one supplying all the tech and knowledge. The Turtles are then forced to go to the surface in order to meet up with Splinter.
In comparison, "Rise of the Turtles" has a slow beginning that features a lot of exposition about the Turtles' background (something that isn't explained in the 2003 series until episode 3) and the Turtles begging Splinter to allow them to go to the surface for their 15th Mutation Day. Leo is shown to be a sci-fi geek, which (in my opinion) is an interesting and rather adorkable touch to his character, but there is generally too much exposition that isn't tied to the narrative. Once they reach the surface, there's a lot of oohing and aahing and eating pizza; again, nothing much happening until they see April being kidnapped by Kraang.
In contrast, in "Things Change," the Turtles get into trouble when they go topside because Raph's impulsiveness gets him locked into an armored van. Again, this is more action tied to narrative and revealing of the Turtles' personalities through the unfolding of the plot. In the process of rescuing him, Mikey is shown to be a wisecracking but effective jokester, Don rescues Raph by hijacking a numbered lock, and Raph's quick temper and belligerence is shown. The episode also introduces the Foot Clan, who become the Turtles' main enemy later on.
I did appreciate the scene in "Rise of the Turtles" where the Turtles keep getting in each others' way during their first fight. But the scene in which the Turtles wait for the van, while establishing Leo as a capable leader, was again incredibly slow-moving. Oddly, the episode almost seems to make out Raph as the protagonist; he's the one who wins the initial sparring match, and he's the one who's the focus of the Turtles' first fight against the Kraang.
Overall, "Things Change" does a better job than "Rise of the Turtles" at both conveying the Turtles' personalities and presenting a narratively coherent and action-packed plot.
"Meet Casey Jones" (2003) & "Turtle Temper" (2012)
These episodes are interesting to compare because both of them deal with the same subject: Raphael's temper.
To be honest, "Turtle Temper" is one of the few episodes of TMNT 2012 that I enjoy as a whole (instead of just for the funny moments). I think it does do a good job at exploring Raph's coming to grips with his temper. It's interesting to note, though, that "Meet Casey Jones" and "Turtle Temper" explore Raph's anger from different angles, which have different implications for Raph's character as a whole.
"Turtle Temper" shows the downside of Raph's anger as interfering with the Turtles' missions and ability to work as a team. However, "Meet Casey Jones" has a much darker take: Raph's anger isn't just an obstruction, it's deadly. In his anger, Raph almost seriously injures Mikey, his brother. These two different premises also result in different reactions of Raph to his temper: 2012 Raph fails to see why his quick temper is a bad thing, while 2003 Raph is genuinely upset about his anger (as evidenced when he ran to the surface and yelled, "What is wrong with me?!"). 2003's take is more intriguing in the sense that in a later episode (3x16 "Darkness Within"), in which each Turtle is confronted by his worst fear, it's shown that Raph's worst fear is himself. Ultimately, this isn't explored further, but it's certainly a very nuanced take on Raph's character. Raph is not afraid of risking his own life or of facing powerful enemies; however, he is afraid of going out of control and hurting others as a result.
"Turtle Temper"'s overarching plot is Raph's realization that his temper can be a bad thing, although he realizes it in a fairly passive way: by listening to a story Splinter tells about himself. "Meet Casey Jones" has a more interesting structure, as Raph comes to grip with his anger by seeing it reflected in Casey Jones. He also apologizes to his brothers afterward and shows genuine regret—something that 2012 Raph doesn't do. It's a small touch, but small touches can make a big difference where characters are concerned.