In both cases you're well protected against the solar wind by distance and atmosphere in the case of the Mars rover and by the Van Allen belts in the case of the Space Shuttle. But both still have to worry about cosmic rays. Mars's thin atmosphere helps a bit with that but not nearly as much as you'd like.

EDIT: I suspect a lot of what lets you use Snapdragons in the rover is the idea of making error resistant systems rather than error resistent devices. The same thing that let us replace a single $10,000 hard driver with ten $100 hard drives in a RAID with the same performance and reliability despite the individual drives being much worse.