This is exceptionally simplistic and reductive
This endless debate. Look if professionals can't figure out exactly what the difference was, than it's not going to be resolved here.
Bottom line. Patricians started with political privileges that slowly eroded over time.
They cannot become a tribune (this is the big reason the distinction matters!) and they make up most of the ruling class, Almost all of them were extremely wealthy. They were vastly outnumbered by the plebeians.
The plebeians include a full range of wealth - from super rich to dirt poor. Because there are obviously more poor than rich plebs and because pretty much all of the patricians are rich, the divide is often reduced to simply money. But it's not just money.
It almost seems like there was an ethnic division in the early history of the Republic, which was reinforced by the banning of intermarriage between the two groups in the 12 tables. Then again, maybe the patricians are simply the former nobility of the Roman Kingdom? Maybe it's both - maybe they were the old nobility and that nobility was ethnically distinct from the plebeians (this is personally my bet). Maybe the patricians were a ruling Etruscan class and the plebeians the indigenous Latins? It's fascinating stuff, but it won't be solved anytime soon.
As the game progresses, the patrician-plebian divided gradually becomes less important and the rich-poor divide more important. Patricians continued to snobbishly lord their pedigree over the plebs, but it increasingly meant less and less in terms of real power. All except for one issue: Tribunes (where plebeians have the power). So I'll say again, we need a cursus honorum, and we need tribunes. Next to the consuls, the tribunes held the most power in the whole government.