Observations from the recent Whole Heliosphere Interval (WHI) solar minimum campaign are compared to last cycle's Whole Sun Month (WSM) to demonstrate that sunspot numbers, while providing a good measure of solar activity, do not provide sufficient information to gauge solar and heliospheric magnetic complexity and its effect at the Earth. The present solar minimum is exceptionally quiet, with sunspot numbers at their lowest in 75 years and solar wind magnetic field strength lower than ever observed. Despite, or perhaps because of, a global weakness in the heliospheric magnetic field, large near-equatorial coronal holes lingered even as the sunspots disappeared. Consequently, for the months surrounding the WHI campaign, strong, long, and recurring high-speed streams in the solar wind intercepted the Earth in contrast to the weaker and more sporadic streams that occurred around the time of last cycle's WSM campaign. In response, geospace and upper atmospheric parameters continued to ring with the periodicities of the solar wind in a manner that was absent last cycle minimum, and the flux of relativistic electrons in the Earth's outer radiation belt was elevated to levels more than three times higher in WHI than in WSM. Such behavior could not have been predicted using sunspot numbers alone, indicating the importance of considering variation within and between solar minima in analyzing and predicting space weather responses at the Earth during solar quiet intervals, as well as in interpreting the Sun's past behavior as preserved in geological and historical records.