The processing power in current exploration rovers like the MER's RAD6000 (20Mhz) and Curiosity's RAD750 (200Mhz) is far below anything you would find in any mobile phone today. As far as I understand this is mainly due to the radiation hardening and possibly temperature ranges.
What would happen if you used terrestrial processors and other electronic components for a rover on a lunar mission? Could you use other measures to prevent these effects (e.g. shielding)?
It is a bit a question of philosophy - between a zero-fault system or a fault-tollerant system. A typical terrestrial processor would cause a higher number of faults during computation, such as bit flips etc. leading to unpredictable results. Yes, one can shield them to some degree. But it is much easier to build the computer and its software so that they can handle such glitches. This is essentially what is done in a number of academic satellite projects. Those projects either just can not effort radiation-hardened hardware or are just not allowed to buy it (dual-use issues). Although academic satellites are mostly sent to low Earth orbit only, they have also an increased problem with radiation and solar activity. Nevertheless, a wide range of cheap embedded and even desktop CPUs have been successfully used.
An alternative to hardened processors is the concept of multiple redundant negotiating processors, where each processor negotiates its view of the world with the others, so if one suffers a bit flip it can be corrected by consensus, or even shut down if it is faulty.
The space shuttle used a basic version of this with 3 computers needing to agree, or if necessary 2 out of 3 winning.
Of course hardening and simplification can provide provable robustness, whereas using a complex configuration with 9 high end processors will have to cope with frequent incidents requiring correction.
