Oh, there were huge difference, just as vast as the differences in social status among freeborn citizens. Some freedmen were as wealthy as anybody in Rome, which was famously parodied in the Satyricon through the character of Trimalchio, the freedmen of a merchant who inherited the estate of his merchant master (it is implied due to a sexual relationship) and became a succesful merchant in his own right. Many others of course did not attain such wealth. Possibly there was a higher floor on how poor a freedmen could be, because if they were truly grindingly impoverished they simply would have stayed slaves, but that is of course an inference from lack of evidence--we don't know about those too poor to leave a material record.

There is a tendency to conflate freedmen with a sort of ancient bourgeois, who lacked the blood to become highly placed in formal political office so pursued their status competition in the economic realm (cf John D'Arms Commerce and Social Standing in Ancient Rome although the argument is older). This is in part because the Roman stereotype of freedmen as being boorish, using wealth to replace native sophistication, and immoral is similar to early modern stereotypes of bourgeois merchants (and, it occurs to me now, modern stereotypes of Jewish people--I wonder if it has something to do with a sense of social untetheredness)(also, of course, the Roman depiction undoubtedly informed the later depictions). Supporting this is a seeming prevalence of freedmen in epigraphic funerary inscriptions, which has then been read to mean that freedmen had a sort of internal cultural milieu, which strengthens the comparison to the early modern bourgeois. Unfortunately this bit of evidence is supported by the probably unjustified assumption that anyone with a Greek element in their name was a freedmen so I'm not a big fan of it.

Ethnicity is a really complicated question here, made even more complicated because by the Roman period "Greek" had become a very vague term referring to pretty much anyone living in the eastern Mediterranean who wasn't Egyptian. One of the required bits of information provided at slave auctions was place of origin, whether the person was from Gaul, Thrace, etc. This provided relevant information for the buyer, who might prefer a Thracian if they wanted a shepherd or a Greek Asian for a teacher, but it is difficult to know how much this would carry over into freedmen status, particularly if one was born a slave.

Incidentally, the father of Horace, arguably the greatest Latin poet, was a freedmen.

There is a recent edited volume called Free at Last! The Impact of Freed Slaves on the Roman Empire which I have greatly enjoyed the bits of that I have read.