In short
The Iguaçu falls stretch nearly three kilometres and are shared between Brazil and Argentina. Both banks are World Heritage sites and are visited separately.
Brazil gives the panorama, in half a day. Argentina gives proximity — the walkways lead out over the Devil's Throat — and needs a full day.
What it really costs
Foz do Iguaçu is a town of no interest in itself and cheap, which makes it the logical base. Puerto Iguazú, on the Argentine side, is smaller and dearer in foreign currency.
Each park has its own entry fee, and both are paid. Doing both banks is the norm, not an extravagance: they do not show the same thing.
What nobody tells you
Crossing from one country to the other is a border crossing with checks. One park's ticket obviously does not cover the other, and taxis charge more for the international run.
The flow of the falls varies enormously. In the dry season it halves; after heavy rain, the Devil's Throat walkways close because of flooding.
The Itaipu dam, twenty minutes away, was the largest in the world before the Three Gorges. It can be visited, and it is the second reason to come here.