There were some class based distinctions. Only senators could wear purple, and there were many laws and restrictions on just how much and where it could be on a garment throughout the years. Senators and some other officials also had a special chair called a curule chair. All the equites, or knights were allowed a gold ring.

The usual indicator of a rich Roman was multiple homes, lavish furniture, lots of books in some cases, fine clothing, imported goods, dinner parties, land speculation, (slum) land-lording, buying votes, and funding even-more-lavish-games-than-that-last-guy-who-was-aedile; really any of the usual conspicuous consumption, demagoguery, land-lording, etc., you can think of. Rome had a number of sumptuary laws to combat this decadence at different times.

Outside the Emperors, Crassus was probably the richest Roman ever. He owned many homes, had a big pile of gold, lots of slaves, and owned a lot of property in Rome that he rent-farmed as a landlord. Supposedly the Parthians poured molten gold into his mouth (possibly after he was killed in battle) to "quench his thirst".

Something that is not mentioned i the other answers here but is genuinely important in the Imperial period is the patron/client relationship. Patronage has existed in the Republic and was a relatively institutionalized form of social structuring, primarily used for political purposes during that era, but after the end of the Republic, it continued on. Patrons would give gifts to their clients, and thus, a large part of a person's day in Imperial Rome was, quite simply, waiting to see your patron. Why would someone who was wealthy just hand out money for nothing? Why, to show that they could of course. There was a video I saw not too long ago with one classicist likening it to modern day yachts, in that one doesn't need the biggest yacht. It is, rather, about rivalry with peers. In this sense, the most important thing about being wealthy was showing people you were wealthy through direct actions. This was a key component of Romanization as well. Most monuments in the Roman Empire during the early Imperial period were not purchased by the Emperor, but rather, the local elite (who, operating on the 'Roman' model, spent on the community and their 'clients'). In this sense, one can characterize the spending of Emperors in Rome as a direct analogy to this 'client-patron' relationship on a much grander scale.