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What are the advantages and disadvantages of using solar power to run probes on Mars? Do those advantages and disadvantages extend to using them here on Earth?

An advantage is that solar power is constant and predictable, but a disadvantage is the Martian distance and atmosphere.

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Solar power is a wonderful form of power. The sun's nuclear fusion reaction is consistent, steady, and likely to last for another 4.5 billion years. That makes solar power a dependable energy source, and it has been powering all kinds of Earth based systems and satellites for a long time. It is proven technology, and that by itself is one big advantage to using solar power to run probes on Mars. We understand the technology, and it has been proven to work. Mars is also relatively close to the sun, so Martian probes that are solar powered benefit from receiving enough sunlight to make solar power a viable power option. Obviously, Mars gets less solar energy than Earth, but it is enough to power Martian probes.

A disadvantage to using solar power on Martian probes is the Martian atmosphere. Probes in orbit around planets or probes that are making their way through the solar system do not worry about atmosphere and related problems. Martian probes that are on the surface are subject to Martian storms and the resulting blowing dust. That dust can cover the solar panels and reduce their ability to gather the necessary solar power to power the probe's various systems. This is generally not a problem on Earth as humans can systematically clean the light collecting surfaces, but as of yet, there isn't a way to effectively monitor and maintain the solar power collecting surfaces on Martian probes.