Why do ethanethiol molecules have stronger London dispersion forces than those of ethanol?
Oooh, interesting question. Let me have a go at this and we'll see if there's someone else who can put it in better terms. 

So, the way I tried to reason this out is to remind myself that every single bond, in each of these two molecules, is the same except the ones where C is bonded to O or S and the ones where H is bonded to O or S. We then pay attention to these bonds, particularly. 

Now, at this point, we also have to remind ourselves that O is more electronegative than S and therefore, pulls electrons towards itself much more than S does and both of these atoms are a tad bit more electronegative than C or H. Therefore, the region of higher electron density in ethanol is O and the region of higher electron density in ethanethiol is S. We also have to remember that O has a more negative partial charge than S does. We also need to remind ourselves that S is larger than O which, by the way, is also a contributing factor as to why O is more electronegative than S. 

I then uses these facts and try to make sure my reasoning is sound; If we look at the way london dispersion forces work, we realize that it is easier for electrons to fluctuate between molecules when they are within the bonds than when they are within the atoms constituting the molecules. In other words, when the electron density is concentrated in the bond, as is the case when there is 0 electronegativity difference, then london dispersion forces are stronger. (Electron fluctuation, by the way, is the basis for dispersion forces) 

Using the reasoning above, it becomes natural to think that the dispersion forces in ethanthiol are stronger, since the electron density is more concentrated in the bonds than it is when you consider ethanol. Therefore, it is easier for the electrons to fluctuate between two ethanethiol molecules than between two ethanol molecules, which is why ethanethiol molecules have stronger dispersion forces between themselves as opposed to the ethanol molecules.

Note: However, ethanol molecules have a much more powerful intermolecular force available to them, which is hydrogen bonding. So, clearly, ethanol molecules have a higher boiling point or whatnot than ethanethiol. Just to make sure that this is clear enough.