*Warning: a big block of text incoming*
The actual background conflict should have had some weight in the characters' actions and motivations. That's one of the biggest flaws in the PT.
In the OT we care about the Rebellion vs Empire civil war because it determines the paths our heroes crosses, how they view their surrounding world and how they think.
The discovery of the Death Star's secret plans, the death of Owen and Beru and meeting Obi-Wan Kenobi makes Luke join the Rebellion, follow his father's steps and leave Tatooine for some wacky and fun adventures across the galaxy. The fate of Alderaan and the menacing power of the Death Star makes Han have a chance of heart.
The destruction of the Death Star brings hope to our heroes, which later determines the themes and scenes of Empire Strikes Back: Han must choose between becoming someone like Boba or a respectable figure like Lando, Luke grows up, the "I am your son", the depressing ending, the Empire being everywhere.
Thing became flatter in Return of the Jedi with characters stopping being themselves, plot canyons, Ewoks and cheap rehashes of the first movie, but nevertheless there's still a sense of danger and urgency in the battle of Endor: the Rebellion could be finally defeated, which means really bad news for Leia and Han, Han is finally working with the Rebels full-time, Luke could fall to the dark side and replace his father, the galaxy is waiting. The death of Vader and the Emperor and the destruction of the second Death Star and the Executor in an isolared planet at the edge of the galaxy is the first step in a reorganization of the superpowers and organizations, with the Rebellion making advances and the Empire losing control.
Nothing of this is in the PT.
It does not matter whether our protagonists are facing Dooku, Grievous, the Trade Federation or evil Darth Jar-Jar Binks, the conflict is superfluous.
After so many years of conflict you would guess it had an impact on Coruscant and its people, but the same views of the big city in TPM appears in RotS too. We never see the wealthy people becoming wealthier and the poor people becoming even more miserable, we never see any building in ruins. There's a big space battle near Coruscant at the beginning of Ep. III, but once old Palpy is saved the battle is dropped, never to be mentioned again in the rest of the movie. Utapau, Kamino, Tatooine... All of these planets remains mostly the same. Meanwhile in the OT Alderaan is destroyed, the Rebel base at Hoth is destroyed, Cloud City is evacuated, and the Rebels are pushed back everytime with few if no advances towards the core of the galaxy, the heart of the Empire.
Anakin does not fall to the dark side because he became depressed at the long and bloody conflict. He fell because crazy Palpy manipulated him with false hopes of saving Padmé and saying bullshit about the Jedi Council. That could have happened with the Republic at peace, with a silent coup d'grace.
Obi-Wan became a flat, boring character, and yet another Jedi rather than the wise crazy wizard he was in Star Wars. The Jedi are all the same, they never ever develop their own personality and traits (although to be honest some traces of this can already be seen in the Luke from RotJ). Why Cound Dooku is working with the Separatists? We never know anything about him other that "he's evil". Who's Grievous? Yet another evil character that does not matter. Vader was the embodiment of all the hate the heroes in the OT had towards the Empire. Things were personal.
What if the war in the PT has a real impact?
Let's say the Separatists have a good cause. The Republic is corrupted, they're oppresing Outer Rim worlds and some systems wants to leave the Republic or oppose the overly centralization of power. But the war is unavoidable after the actions of the Republic in Episode I.
The Separarists, exploiting the lack of a republic army and the corruption and incompetence of the Republic leaders, are making some real advances and causing great losses to the Republic. Coruscant is either in ruins or no longer safe, so the Senate and the main organizations are evacuated. Things become tense inside the Jedi Order, with opposing factions hotly debating how should the Republic act and what actions should the Jedi follow.
Obi-Wan and Anakin are disappointed but they also have opposing views. Anakin chooses order and eficiency over freedom: he has personally seen the great damages the war has caused. Maybe his homeworld ("not-Tatooine") has been ruined, our his family ("not-Shmi") is dead, or he has lost his hands and legs in many battles and he relies more and more in mechanical parts, in a foreshadowing of the man he will become. He sees how the Senate is a bunch of incompetents, how the Jedi are unable to protect inocent people like his family, and how the Jedi ideals of justice and peace are a big fraud. He sees himself as a highly competent and skilled Jedi and commander, he instantly connects with Palpy and he's enchanted with the chancellor's promises of power and peace. For the greater good.
Obi-Wan and Dooku are mostly similar: they haven't had any personal loses, so they cannot understand how Anakin feels. However, Obi-Wan works inside the Jedi Order and the Republic, while Dooku left the Order years ago (when he understood the Order was beyond repair) and started working on its own. So both are nice and good people ("from a certain point of view"), working for the same cause, but they doesn't know.
The Republic only starts striking back, reconquering lost worlds, restarting the economy and guarantying the security of its citizens when Palpy, with Anakin at his side, makes morally questionable moves. That polarizes the Jedi: some believe the old order will be restored at the end of the war so it's ok to support Palpatine's actions, some follow Dooku's steps, and some actively opposes the government.
The Separatists also became desperate in their attempts of winning the war, which further increases the chancellor's public approval.
At the end the Republic wins and it's finally corrupted, the remains of the Separatist cause are reformed into the Rebellion, the Jedi are considered enemies of the Republic (thus killed, with the exception of Obi-Wan and Yoda). The militarization of the Empire continues, and the arrival of the Star Destroyers marks the first time the galaxy sees ships so massive and big. The Empire is everywhere.
This way you can remove or restore original ideas from earlier drafts/stories, or introduce new twists:
-The Republic/Empire, rather than evil, was corrupted beyond repair. There's no Sith conspiracy, no prophecy, no Sith order: just the wrong people in the wrong places.
-Rather than star systems fearing openly joining the Rebellion, they actively oppose then: the Separatists were reformed into the Rebellion, and the Separatists causes many great losses during the Clone Wars, so why opposing those who bring peace to the galaxy? While the consequences of the war in the PT could be seen in the core worlds, the deadly battles in the OT are at the edges of the galaxy, too far away to create a political and emotional reaction in Coruscant and nearby systems.
That would have nice consequences for the OT:
-Obi-Wan's exile isn't just for escaping the Empire's sinister hands. It's also a form of self-punishment, and a way to meditate and reorganize his ideas. He understands he was wrong all this time, and that his failures contributed to the creation of the Empire. He's hesitant at training Luke, up until he watches Leia's recorded message and understands he has no option. That also makes his sacrifice in Star Wars more interesting.
-Owen and Beru, like many people across the galaxy, opposed the Jedi and the old Republic by the end of the Clone Wars, which ironically makes Luke more and more rebellious towards the establisement.
-The heroes in the OT aren't just fighting against the Empire. They're facing the whole galaxy, making the end of all the OT movies a more satisfying experience, and the universe a bigger place, rather than the shriking we had with the actual PT.
-Most of RotJ's faults could be redeemed: Luke's reaction at Obi-Wan's "from a certain point of view" is still weak and lazy, but thanks to this fixed PT both the line and Obi-Wan's actions in the OT have a new meaning. You could understand the passivity of Yoda, Vader's redeem is more believable, the Emperor is again the corrupted and efficient leader rather than a crazy old man.
-You would have a nice mix of the adventures and actions of Star Wars and the complex character development of Empire Strikes Back, so both public would be happy.
-The evolution of the technology in the Star Wars galaxy finally have sense: the droids doesn't suddenly dissapear from the story, the ships in the PT are more primitive, the Star Destroyers are the high-end, latest advance in military fields, and so on.
