Well first thing is the average 'citizen'. Outside of Italy which received citizenship after the Social War in the Late Republic, most of the people living in the Empire weren't citizens until Caracalla passed a law in 212 CE. Now you had pockets of citizens, usually concentrated in cities made up of the provincial elites, or veteran colonies, or the odd city given citizenship. But the vast majority of people in the empire weren't citizens until the early 3rd CE.

That sidetrack aside, it depends on which 'Republic' you're referring to because the Middle Republic is very different from the Late Republic. This also applies to the Empire, the early Principate is significantly different to the 3rd Century Crisis which differs to the Dominate which differs to the Late Empire.

I think I can safely say that the Middle Republic and the early Principate was definitely better than the Late Republic. By that time the Republican system was at the end of its tether and political violence and unrest pretty endemic. There were the allied states who felt denied their fair share of Rome's success, the disenfranchised from the various power struggles, the poor who'd been migrating from their farms to the city (the economic issue is itself a big thing), and very ambitious nobility.

Now the civil wars wouldn't directly target your average citizen - proscriptions and political murders would be targeted at other nobles or the wealthy equestrians. But rampaging armies weren't terribly respectful of personal property and the dislocation of infrastructure caused by wars would have serious affects on people's subsistence living. So this was a bad period and partially why Augustus was so beloved because he brought stability and peace.

Now quality of life-wise, I'm not certain. Economic policies and cheap slaves had seen a dwindling of independent small holders who drifted into the cities looking for work. You had more tenant farmers renting land from rich landowners. Despite the cities' increasing luxuries (e.g. baths, public entertainments), simply being in a city was an easy way to shorten your life-span due to things like the rampant poverty, crime, and disease experienced by the poor. While Rome and some cities had the corn dole, it wasn't meant as a welfare scheme - it was a reward for citizenship and that meant the wealthy freely took part in it to defray the costs of their household. And they had the connections to make sure they were on the dole (and ensure they didn't get trimmed by Augustus when he revised the lists).

Militarily, the Empire was better off. It regularized and reduced the legions to an number the state could afford (Augustus had to do some mass layoffs after his defeat of Anthony). The ending of the wars of conquest made the political situation more stable since there were less opportunities for nobles to build up sufficient reputation to make a play for the throne (which still happened with weak/unpopular rulers). The regularization of the auxilia helped spread Romanization beyond the borders increasing security somewhat. Note that Rome had always used auxilia previously to fill out tactical gaps (i.e. cavalry, missile troops, skirmishers) but this time it was made permanent.

As for internal unrest, I've touched upon it already but politically it depended on if the emperor was popular (especially with the soldiers). Weak rulers would invite challenges from generals on the periphery with an army and sufficient reputation to bring the soldiers on his side - consider 69 CE with Nero and the resulting 4 different usurpers.

Rome itself was also a potential powderkeg. Emperors knew that they were living alongside 900,000 citizens who were mostly living a hand-to-mouth existence. Keeping the population somewhat content was very important and for the most part they succeeded in doing so.

I can really only speak in regards to the military capabilities of the late Republic and the Empire. Now obviously we must take into consideration that although we often think of ancient Rome as being one consistent entity throughout its entire existence, the Romans themselves were often just "citizens". By this I mean, your average roman citizen was still tending to the fields, fishing, or just living day to day life, mostly unaffected by the overall politics in the capital in both the Republic and the Empire. Excluding any technological differences, you would likely have a hard time knowing which era you were in if you were in an average roman village. In terms of it's military capacity, the Empire would have a strong edge. The romans adopted and tweaked their tactics and orginization over many years of brutal fighting and countless phyrric victories and devastating losses, learning from each experience. This alone made the armies of the Emperors extremely effective when well commanded. The empire brought with it larger legions, more legions, better auxillary units, and most importantly more men. The early and mid republican armies relied heavily on older "middle class" and Noble recruits for it's infantry. These men often paid for their own equipment. While this does give well equipped and experienced men, it also limits the size of the legions, as there were only so many men in a position to join. The emperors army on the other hand tended to appeal to anyone that was looking for a place in life and a grant of land. This gave the later legions a near endless pool of manpower. This on top of the auxiliary units the empire provided and the wealth of experience in the average Roman legions gives the Emperors army a sharp edge over the Republican.

I hope that this answers at least part of your question! Apologies for the lack of formatting as I am on mobile.