A condominium unit is a separate piece of property that can generally be bought or sold without association approval, with its own property tax bill, its own mortgage, and so on, with some common maintenance and covenant enforcement vested in a home owner's association in most cases.
A co-operative, mostly found in New York was a pre-condominium way for people to live in units of multi-family buildings while having an ownership interest. But, in the most pertinent difference from a condominium they don't, or at least, historically didn't, have separate mortgages for each unit. The whole building had a common mortgage, so the co-operative had an intense interest in screening would be members for long term creditworthiness and general personal reputation, which is an insidious affair, especially when conducted by your neighbors. Also, once granted control over whom one's neighbors are, there is a natural tendency of co-operatives to regulate who can buy into a co-op, not simply to insure creditworthiness, but also to manipulate the prestige of the co-op and with it, the value of the properties in it. It is a club in which membership is more valuable when it is more strictly limited.
A recent discrimination lawsuit at one of the most high end co-ops in New York City, the Dakota, illustrates just how much of a mess this approach, which invites wide ranging meddling into the character and personalities of one's neighbors, can produce, although the New York Times article does a poor job of explaining how the differences in history and legal organization of co-operatives from condominiums is pivotal in creating the kind of situation described.
One irony is that in this and in a wide array of private law areas, New York's very late adoption of "no fault divorce" being another, the center of lassiez-faire capitalism and private enterprise is among the most backward and byzantine in the United States.