A condominium is completely different from a leasehold. Perhaps most importantly, a leasehold is for a fixed term of years. That means that whoever improved the property understood that the resale value on their improvements would be less; increasingly so as the end of the lease approached. It's basically not much different than any other kind of lease, such as a lease of an apartment. In both cases there are rules that control, for example, how a landlord must compensate you for improvements to the property (i.e. building a house on the land, or modernizing the wiring of an old apartment) that he would benefit from after the termination of the lease.
Residential leaseholds are less common in the United States than in Common Wealth countries because of the different political, legal, and housing development histories. But they certainly exist in the U.S.
By contrast, each condominium in a building is usually held in fee simple--perpetual, absolute ownership--except that there are always a common set easements and servitudes related to the obvious fact that each condominium is intrinsically and substantially reliant on the other condominiums--access to supporting structures, plumbing and electrical risers, etc. Shared spaces are held in common by all the condominiums, and title in those shared spaces can't (for obvious reasons) be severed from the title in your condominium. In other words, you can't sell your condominium but keep your title to the swimming pool, even though both titles are fundamentally distinct. Furthermore, on top of all of this are contracts that mediate how the interdependent obligations are to be carried out. Basically, a condominium is a pre-packaged set of titles and contracts that both legislation and courts recognize as a holistic, interdependent set of property rights when mediating disputes. Leaseholds, by contrast, are far simpler.
Sometimes you can have a situation where a condominium is constructed on leased land. There's a huge residential condominium in San Francisco's Chinatown like this. These condominiums sell for substantially less than market prices because, once the lease is up, there's no telling how much the land rent will increase or if there might be a forced sale of the condominium to the land owner. I imagine there are lots of condominiums like this across the country. It seems like a good way for a developer to continue to extract rent while maximizing an upfront return on investment.
OTOH, buyers are especially wary of purchasing homes utilizing unfamiliar legal instruments; cities are likewise wary of permitting such developments; and banks often won't finance the purchase of such units. So these kinds of properties sell at more of a discount than you would theoretically expect, especially in the United States.
There's another kind of shared ownership of residential buildings called tenancy in common (TIC). Usually you only see this happen when a deceased landlord's estate is passed to multiple heirs. Because San Francisco restricts the number of condominium conversions that can occur each year while simultaneously restricting new developments, developers began doing TIC conversions as an end-run around city planners. As these became more common and more marketable, some local banks began to finance the purchase of these units. AFAIK, San Francisco is the only housing market in the country where you can get a mortgage from a bank to purchase a TIC unit without any other form of security. They're a good option for home ownership in the city. There's some additional legal risk, but my guess is that compared to the typical hassles you see with small condominiums, worth it given the discount.