[antognini]

A diffuse reflector like the moon behaves differently than a specular reflector like an ordinary mirror. A specular reflector will preserve the image but a diffuse reflector will destroy it.

To get a little technical, if the Sun is reflected by either a diffuse or specular reflector, the integrated surface brightness over 4 pi steradians surrounding the reflector will be equal to the integrated surface brightness around the same point if the reflector had not been present.

The difference is that in the case of the diffuse reflector, the specific intensity becomes isotropic, but in the case of a specular reflector, the specific intensity is unidirectional. Now, from your perspective somewhere off to the side looking at the reflector, you only intercept a small fraction of the magnitude of the specific intensity in the case of a diffuse reflector. But you intercept the entire magnitude of the specific intensity if the alignment is right, and zero otherwise. So the surface brightness you're able to observe is unchanged in the case of a specular reflector (if the alignment is right), but is reduced by the solid angle subtended by the reflector as seen by the source divided by 4 pi.

So it is not surprising that if you take a magnifying glass to an image of the Sun in a mirror you can start a fire. But this doesn't mean that you'd be able to use a diffuse reflector like the moon to do the same thing. After all, a white baseball is also a diffuse reflector and is reflecting the Sun's light on a sunny day. But I doubt you will be able to put a magnifying glass to it and light a piece of paper on fire. But you should get a big Fresnel lens and go out the next time there's a full moon and prove it to yourself! (The big Fresnel lens will be fun for a lot of other experiments, too.)