Why can't light pass through a gold foil, but alpha particles can?
Light can pass through a gold foil though, it just has to be thin enough. 

Pure gold is a very malleable substance and can be beaten with a hammer into foils of around 100 nm thickness. Sources suggest the gold foil used in the Geiger–Marsden experiment (known more commonly as the Rutherford gold foil experiment) was about 86 nm thick. Somewhere around this region of thickness, gold starts letting light through, as well as alpha particles. It is well known that ~20 nm thick gold (or any other metal) deposited onto a surface by thermal evaporation in a vacuum is essentially invisible and can be used as a transparent conducting surface/electrode. As the foil gets thicker, the gold starts to absorb a noticeable amount of photons, especially in the violet/blue part of the electromagnetic spectrum, while scattering away the red light. The light scattered off a semi-transparent gold film acquires a reddish/yellow tinge, while the light that still does make it through is blueish. When the film reaches 100 nm or so, it becomes practically as opaque as the bulk metal.

In truth, comparing the mean free path or collision cross-sections of photons and alpha particles through a gold sheet is somewhat more complicated than just saying alpha particles should penetrate less because they're massive, so it would be conceivable for a film of appropriate thickness to block photons yet still allow alpha particles through. A very coarse, hand-waving explanation is that photons are absorbed in the electrosphere of a gold atom, while an alpha particle is only significantly pushed away if it bangs almost straight into the nucleus, a region composing approximately $10^{-15}$ of the volume of the atom.