The Brazilian side of the falls — accessed via the Tancredo Neves Bridge to Foz do Iguaçu — gives you the panoramic, postcard view of what you've been standing inside of for two days. The Brazilian park is smaller, the visit shorter (half a day is plenty), and the angles are completely different. A taxi crossing costs about USD 25 each way; immigration is straightforward for most nationalities.
The Wanda gemstone mines, an hour south on Ruta 12, sit on top of a layer of amethyst-rich basalt and offer mine tours followed by a slightly hard-sell visit to the shop. The drive there cuts through some of the most striking red-clay landscape in Misiones.
For something more substantial, the Jesuit Missions ruins at San Ignacio Miní lie three hours south and are well worth an overnight — a series of 17th-century mission settlements built by the Society of Jesus among the Guaraní, now a UNESCO site. The route is best done as a loop with a stay in Posadas.