What are the differences in design considerations between Moon rovers and Mars rovers?
Power: The Moon has a night that lasts for 14 Earth days, and it gets really cold during the night. This makes it difficult/expensive to design a rover that can last through the night: you really need a radioisotope power source (RTG) to provide at least heat overnight.
Mars, on the other hand has a day/night cycle of slightly more than 24 hours, so solar-powered rovers are feasible.
Communications: On the near side of the Moon, communications with Earth is easy because Earth is nearby and always in the same region of the sky.
Mars is a lot further away, so Mars rovers mostly use a communications relay (a satellite in Mars orbit).
Atmosphere: The Moon is a vacuum. This makes life difficult: lubricants evaporate, excess heat has to be radiated away. And due to the lack of wind, Moon dust is really abrasive.
Mars has a thin atmosphere. Heat management is a bit easier, although it does mean the rover cools down quicker so more heating is needed. Dust storms are common and can endanger the mission by covering your solar panels/blocking the light.
Landing: With its low gravity and no atmosphere, the Moon is easy to land on using rockets.
Mars' atmosphere is too thin to provide all the aerodynamic braking you need, so you get a complex sequence that starts with a heat shield, then a parachute and finally a rocket-powered descent.
Gravity: In both cases, gravity is a lot lower than on Earth. Rover designers take advantage of this by designing a lighter rover, with structures that would be too weak to support the rover on Earth. Because the difference in gravity (Mars's gravity is about twice that of the Moon) you can't take a rover intended for the Moon and just drop it on Mars. But the general principle is the same, so a rover design for one could be adapted easily for the other.
Day length: As stated above, the moon has a night which lasts 14 days, while Mars has a day/night cycle of about 24 hours, 37 minutes. This has several implications.
Power supply: although the Mars day is more similar to that of Earth, and solar power is more of an option, Mars is also further away, meaning the irradiance on the martian surface is less than that for Earth/the moon. This means that a solar powered robot designed for the moon would have less available power on Mars, due to there being less sunlight.
Atmosphere: this is related to point 1, as more atmosphere on Mars attenuates some of the available light, further reducing the solar panel efficiency.
Temperature: without an atmosphere on the moon, the lunar nights are very cold. Similarly, the (thin) atmosphere on Mars does tend to trap some of the heat of the day, so the shorter nights on Mars aren't as cold as on the Moon. Practically, this means that a solar powered robot may just be able to power down for the night, while one on the moon may have some components (bearing lubricants, battery electrolytes) which freeze. While freezing itself may be less of an issue in some cases (Teflon bearings instead of grease-filled, for example), as soon as the sun begins to rise over the robot on the Moon, it instantly begins to warm up. This can cause thermal shock in some parts, and the cyclic heating and cooling effects over time can cause further damage.
Gravity: There is less of it on the moon, and the moon is also much closer to Earth. This means bigger payloads to the moon are possible, as the velocities (delta-V) needed to escape Earth's gravity are lower for a trip to the moon. Practically, this means less fuel is used, and so more payload of the rocket can be taken up with the robot itself.
There is a simple comparison chart of the total distances that both Mars and the moon robots have travelled on Wikipedia. Specifically, this image:
As a very brief comparison between the Opportunity (on Mars) and the Lunokhod 2 (on the moon), Opportunity weighed 1063Kg, was solar powered (it hibernated at night/during dust storms). It ceased to function after a dust storm likely covered the solar panels.
The Lunokhod 2 weighed 840Kg, and was again solar powered. At night the solar panels closed up like a clam-shell to conserve heat, and the robot was fitted with a radio-isotope heater to keep the electronics warm. During driving, the open lid with solar panels) touched the side of a crater, picking up dust. When the lid was closed at night, this dust was dumped onto the radiators. As dust is a good insulator, the robot eventually overheated.
Vacuum on the Moon creates big problems for heat removal. And also for friction surfaces. Another issue is the extreme temperature conditions during a long moonlit night and lunar day.