The most influential argument on this topic regarding political theory is Benjamin Constant's distinction between ancient and modern liberty. In short, ancient freedom as practised for example in the Roman Republic was about collective self-determination. Modern freedom, on the other hand, as practised in modern republics, is dependent upon safeguarding the liberty of the people through the constitution. In practice, this distinction is somewhat blurry as the Roman Republic had safeguards as well as a constitution- but what Constant is getting at is the scope of these in the modern republic.

Institutionally, as well, the Roman Senate is incomparable because the authority which they had was not codified. Such authority would be incompatible with the standard modern notion of republicanism.

One major difference with the Roman republic was that it was designed to be undemocratic. Within the comitia centuriata, the assembly that voted on elections of high magistrates (consuls, praetors) and declared war and peace, the population was organized into 193 voting 'centuries'.

Each century was comprised of men within a certain property class (e.g. wealth) and each century had 1 vote - determined by a simple majority within the century. As you can imagine, this meant a wealthy Roman's vote was worth more since he'd be in a century with a couple hundred others while the poor's vote would be diluted as he would be lumped into a century with thousands. To put this in perspective, all of the urban poor were grouped into just 5 (out of 193) centuries.

During elections, the centuries would vote in order of wealth: the centuries from the highest property classes on down to the lowest. Voting would proceed until a majority had been reached. This meant that the poor often never got a chance to vote as the issue would have been decided long before their turn to vote came up. This was of course intentional as the Roman elite detested democracy which they felt was little better than mob rule.

There were other assemblies, the tribal and plebeian assembly which organized citizens into 35 tribes and these voted in random order - making this much more democratic. The latter was used to elect plebeian officers such as the plebeian aedile and the tribunes of the plebs. During the late Republic, the tribunes began putting forward legislation to the plebeian assembly which was binding on the entire republic. Formerly, senate approval was sought for such legislation but in that fractious time ambitious tribunes frustrated with the senate's gridlock just ignored the senate and went ahead - which only contributed to the instability and violence.

A final comment - the division of the population into patrician and plebeian families was no longer significant from about the middle republic. By then the plebeians had won the right to stand for higher offices and slowly gained equal access to political power (only a few priestly offices by tradition were limited to patricians). By the late republic you could have filthy rich and powerful plebeians (e.g. Crassus) just as you could have patricians who'd fallen on hard times and were no longer part of the Roman elite.