Actually, "Nous sommes riches" is a more formal translation of the English sentence "We are rich", which means, you are more likely to see it in poetry and literature, while you would rarely hear "nous" in everyday French. Likewise, "on" (which, in case you haven't heard, takes the same conjugation as "il" and "elle" (the masculine and feminine third-person singular) and that's why it's "On est riche" instead of "On sommes riche(s)") is a more informal form of "we", so if you're in Paris you're more likely to hear "on" in everyday language. Regarding meaning, both sentences mean the same thing (if "on" is put into the right context, which you can determine with other sentences around it in spoken language). So basically, there is almost no difference between the two except formality and grammar.
Hope I answered your question! :)
Je suis = I am Tu es = You are (singular) Il est = He is Elle est = She is Nous sommes = We are Vous êtes = You are (plural or formal singular) Ils sont = They are
"On est" is a little bit special. It is always use with the same form as "il" et "elle" though. Even though more often than not it's an unformal way of saying "we".
Just to make it easy for me from my Spanish perspectine I will think of "On" as "Uno" which means "One" in English: "Uno es rico!"-Spanish "One is rich!"-English "On est riche!"-French Uno, one and on are singular but can be used in a situation that refers to a group of people like: "A place where one can become rich!" I don't know much about grammar but this makes sense to me.
I believe, if "on" refers to "we", the adjective should be plural.
L'adjectif ou le participe passé attribut de on prend le genre et le nombre du sujet que ce pronom représente. Larousse
When “on” means “nous”, the adjective will be plural
That is a sure thing. It may be plural masculine or feminine, depending on who “on” replaces. French Today (My highlight)
« Est-ce qu’on est riches ? » Le Monde
On est riches ! Par la Compagnie Le Théâtre Des Trois Mots Lille la nuit
Alors, on est foutues ? Glamour
So, shouldn't it be "On est riches !"?