Is photosynthesis more efficient than solar panels? If so, by how much?
Rather than considering quantum efficiencies or such details it's instructive to step back and take a broader view. One of the main fuel crops grown in the UK is miscanthus. There are various figures around for the yield produced by miscanthus, but these people estimate it as about 14 tonnes per hectare per year. The energy content is 19GJ/tonne, so that's 266GJ per hectare per year or about 8.5kW per hectare.
Commercial PV panel installations typically produce 500kW per hectare (NB the link is a PDF) though this is peak power and would be a lot less averaged over the year. However, even if averaging over the year reduces the yield by a factor of 6 this still leaves the PV panels producing ten times as much power as miscanthus per hectare.
For comparison the intensity of sunlight at midday is around 10MW per hectare.
Incidentally, I don't mean to belittle miscanthus. PV panels are vastly more expensive to make than miscanthus is to grow, and we have yet to persuade PV panels to reproduce themselves. Both have their role to play in supplying energy.
Photosynthesis is less efficient than solar panels. According to the Wikipedia page on photosynthetic efficiency, typical plants have a radiant energy to chemical energy conversion efficiency between 0.1% and 2%.
Most commercially available solar panels have more than 10 times this efficiency.
The big difference is that plants are "storing" the energy, and solar cells are not. The utility of stored energy is much greater than the output of something like a solar cell.