- You say you stood up and cheered when Leia grabbed a blaster from Luke and shot down the Death Star hallway before ordering Han & Co. into the garbage disposal? Well, I stood up and cheered when Padmé Amidala planned the successful incursion into Theed in TPM, and I cheered even more loudly when she took an active role in that plan's execution. [*snipped for length*]
Those are just three examples, but I could go on. For almost every point supporting the OT, I can find a counterpoint for the PT.
In the comments, plenty of people mentioned how the trilogies complement each other, and having both enriches our experience of the movies. Add that to the compare-and-contrast idea, and you get some of my favorite parts, where the trilogies echo each other and make everything more awesome. I could name tons of these, but I'll focus on two sets of characters: Padmé and Leia, and Anakin and Han.
Padmé and Leia
I love how Padmé and Leia have similarities while still being distinct people shaped by different experiences. You can even see this in their clothes - Padmé has time for elaborate, hand-crafted gowns, while Leia sticks with basic patterns.
They both can aim a blaster as well as anyone - and are willing to do so - and they're both politicians. But Padmé comes from a time where politicians aren't just puppets/targets and can do actual politicking, so we get to see her talking about elections and votes of no confidence and who supports whom or which motion. Leia comes from a time where just surviving is a victory, and the Senate has been dissolved. Padmé initially seeks peaceful solutions (diplomacy in TPM, no military creation act in AotC, the Petition of 2000 in RotS), then turns to violence when that doesn't work - somewhat reluctantly, but she doesn't seem as troubled by it as one might expect given her earlier protestations. (Maybe she realizes how easy violence comes to some people, including her, and that's one of the reasons she tries so hard to put it off?) Leia comes from a peaceful planet, but if she ever believed diplomacy could help against the Empire, she gave up on that a while ago. Their bits of military leadership differ too. Padmé realizes that she can plan a battle in TPM, makes an alliance, and comes up with a three-front strategy; Leia is mostly stuck trying blast a few people while running away - her biggest military planning scene that I can remember is when she gives Rebel pilots the plan for retreat in ESB.
As for personality, they're both adaptable, tough but compassionate, and dedicated to public service, though Padmé talks more about abstract ideals like democracy (maybe Leia just doesn't have the time? or takes it as a given?). But Leia is a lot more bossy, brusque, and sometimes downright rude ("walking carpet"), and lets her anger out more, while Padmé seems to be trying to hold herself in. Again, this partly reflects their times - Padmé wouldn't get far in the Senate by calling senators names (though she will get bold enough to shout at them all to "wake up"), while being calm and non-threatening probably would get Leia nowhere with Imperials. (I also suspect that she internalized some of the Empire's species prejudice despite her conscious beliefs.) And maybe Padmé's background as an elected queen rather than as hereditary royalty like Leia makes her more aware of others' perspectives and less, well, snooty. Of course, different temperaments could play a role here too - Leia shares some of Anakin's impulsive anger. Padmé often denies problems exist until it's too late, such as when she barely blinks after Anakin confesses to slaughtering Tusken children and when she doesn't recognize how shady Palpatine and the Republic have gotten until a few days before Empirization. And of course there's the whole secret marriage thing... It's harder to tell with Leia, but she might notice problems too much, if we read Han/Leia as Lucas apparently intended it, with Leia needing to soften up and realize she needs other people even if those people are not perfect. Which does have some truth to it, but I think it's more complicated than that :D And of course, Leia does suggest that Luke run away from Vader in RotJ, so she isn't entirely free of that impulse.
Anakin and Han
They're both impulsive, love flying (and terrifying their passengers with REALLY SCARY STUNTS), and have some anger and violence issues. Han blasting Greedo (first. ahem.) was effective, but also totally cold and callous, and he didn't seem to have a problem with it. It's no kiddie-slaughter, but I don't think being as comfortable with just... ending a whole life like that as with running to the store for milk is a good sign, you know? On the other hand, seeing Alderaan completely destroyed freaks Han out, and I don't see any signs that he'd be comfortable perpetrating genocide. And he never (physically) injures the people he cares about. More on that in a moment.
They can both be a little pushy around the women they like, too, but in different ways. Anakin is non-physically pushy in expressing how he feels, and he stares at Padmé a little too much and doesn't really seem bothered that it makes her uncomfortable (judging by his little smile when she tells him to stop it). Han is non-physically pushy in trying to get Leia to say she cares about him while keeping his own feelings under wraps, and physically backs her into a corner and kisses her. (Which is dubiously consensual, but at least isn't fatal.)
Now, if it stopped there, I'd say Han is way more bad news than Anakin, but there is that slaughter thing... and that choking thing. Here's where I get into specific scene comparisons. This is one of my FAVORITE trilogy mirror-moments.
Both men have a moment where they see their girlfriends secretly with a Jedi. Anakin yells, "What have you and she been up to?" and then turns his anger on his beloved wife, choking her into unconsciousness and probably contributing to her death. Han waits until Luke is gone, then asks Leia, "What's going on?" When she won't tell him, his first instinct is to walk away. Then he stops, reconsiders, and turns back - possibly realizing that he doesn't know what is going on (even if he know what it looks like), and that more importantly, she's more upset than he's ever seen her. So he apologizes and hugs her. He waits until a better moment to ask if he's right and she loves Luke. When she says yes, he is angry - but he doesn't turn it on her. Again, his response is to walk away. My reading of this is that while Han might shout and grumble, or use his blaster to solve problems involving people who are or might soon be shooting at him, he realizes that you can't force someone to love you. And he's also more open to the idea that what makes him happy may not be the best thing for everyone, and you can't always get what you want - and the galaxy won't end if you don't. And maybe just having a decade or so on Anakin has taught him to restrain his impulses a bit. Anyway, in the end, Anakin turns his anger on the people he loves, and Han doesn't.
I find this a bit more satisfying than comparing the Padmé and Leia arcs, because we see clearly that the next generation has resolved this problem, whereas Leia's arc is so sketchy that we can't be sure if she's resolved... well, what would she even resolve? She hasn't had to face her boyfriend turning into a glowy-eyed evil murderer or seeing her cause corrupted from within, so we don't know how she'd handle that. The only big secret she has to keep that might cause problems is something she finds out toward the end, so we don't know if Secret Evil Father will be as much of a problem as Secret Marriage. The problems with political corruption are only about to start, once the Alliance stops being a hodgepodge driven by necessity and desperation and tries to be an actual functioning government. (Will they be stuck with allies they'd rather not have? Will they try to recreate the Republic as it was or improve on the design? What are their goals, anyway, besides "destroy Empire"? Who knows?) And given that she does have a bit of that ostrich impulse that I mentioned above, it's entirely possible that she could repeat her mother's mistakes. I'm sure there's a way to tweak things slightly, just a teeny bit, a few lines here and there, that would fix this and make everything snap into focus... Any ideas out there?
But overall, I love the way comparing the trilogies like this makes both of them better. And I haven't even talked about Anakin and Luke, or Obi-Wan and Han, or young Obi-Wan and old Obi-Wan! Another day :D