The similarity of the names of these two approaches leads to some confusion. We'll try and sort out the differences.
PASSIVHAUS, often called Passive House in the US, refers to a performance standard for thermal efficiency in a building with the goal of drastically reducing its needs for external energy inputs - 'a home you can heat with a hair-dryer'. It was developed first in Germany where the earliest examples were completed in 1990: since then thousands of examples have been built worldwide. In origin it's a heating-climate strategy and so far few Passive House homes have been built in the American South, though two were completed in Chapel Hill in 2011. The standard does not specify particular construction techniques but rather establishes a set of performance values for energy use which are far more rigorous than those set by most building codes. To enable designers to reach those standards the Passivhaus Institute has developed software tools which precisely calculate the consequence of different design choices as the building design is developed.
While praising the the low energy cost-in-use of Passivhaus buildings, critics contend it is not necessarily the greenest way of building, citing the high environmental footprint of the petrochemical foam products often used in large quantities in Passivhaus construction.
PASSIVE SOLAR is a construction technology which has been around for many years, some would say for millennia. Instead of relying just on improved insulation, passive solar techniques also strive to optimize free energy inputs from the sun. Examples of traditional passive solar strategies can be recognised in the south facing porches common to many North Carolina farmhouses: these provided a warm place to sit and do household chores on a sunny winter day while in the heat of summer protecting the interior from the direct rays of the sun. In the twentieth century these strategies were developed into sun spaces with arrays of south facing windows, essentially bringing the porch space inside, with generous roof overhangs or other shading devices calculated to keep the heat out in summer. This allowed the concept to develop of heating the entirety of the home with the sun's rays, not just a porch or isolated sun room.
Direct solar heating of a home naturally ceases at sunset. Techniques for storing daytime heat for night-time use include increasing the thermal mass of the structure for thermal storage as well as improving the insulation of the enclosure. Passive solar designs in the 1970's and 1980's typically went to thicker, better insulated walls and mass concrete floors as well as earth berms and semi-subterranean 'earth-sheltered' homes. Some of the homes designed in this way performed very well, though many, especially in the South, suffered from summer overheating problems.
While some of the resulting house forms were too eccentric to enter the homebuilding mainstream many significant lessons from this period have been incorporated into the basic thinking of responsible designers and energy-aware builders today. Sad to say this is not typically true of tract home developers who continue to throw down code-minimum homes with no regard for orientation and solar exposure.
In summary then, Passivhaus relies primarily on a super-insulated supertight enclosure to achieve its low-energy objectives while passive solar seeks to maximize the free solar contribution to the same effect. There's a convergence point for these two different approaches: many passivhaus projects include elements of traditional passive solar design. There's also a frequent overlap of each of these strategies with homes designed for so-called Net-zero performance.