Does photosynthesis occur on cloudy days or does it require direct sunlight?
Yes, photosynthesis does occur on cloudy days where the light exposure is diffuse - but the overall picture is not as simple as your solar cell analogy.

Consider one of the findings of the following article: "Impact of clear and cloudy sky conditions on the vertical distribution of photosynthetic CO₂ uptake within a spruce canopy" (Urban et al. 2011), where one of the findings is that


  that shoots from deep within the canopy contribute substantially to the overall carbon balance during cloudy days. But, although shade-adapted shoots had a markedly positive carbon balance over a 24-h period on cloudy days, their performance was impaired on sunny days contributing only a marginal or even negative carbon balance from the middle and shaded parts of the canopy.


So, while leaves at the top of the canopy have a reduced rate of photosynthesis on cloudy days, leaves within the canopy and those that are shade-adapted, still photosynthesised.

I hope this helps a bit.