From George Riley (Central Foundation Boys' School)
I would use sound to tell the difference. In the hollow object there will either be gas or a vacuum (doesn't say in question) but in both of sound travels worse than through a solid. Because the gas particles are further away than solid particles, the particles have to spend more of the energy on movement, to knock nearby atoms along (the very nature of sound). And of course sound can't travel at all in vacuum because there are no atoms whatsoever. Therefore I would make a knock or some sort of sound on one end of the objects, and a piece of sound measuring equipment on the other. The sound will be louder and clear through the solid, and quieter and less clear through the hollow sphere. I think.
Comment from James Handscombe: This is an ingenious idea which probably fails in practice because of the size of things (the Royal Institution Christmas Lectures in 2010 were all about how the size of things affects their Physics: http://richannel.org/christmas-lectures/2010/2010-mark-miodownik. Make sure you catch this year's lectures if you can.) Roughly speaking, the sound will behave differently in the two spheres if its wavelength is small compared with the thickness of the shell. The wavelength of audible sound is between 17m and 17mm and so almost too large (unless the spheres were very large or the sound extremely high pitched). With higher frequency sound this plan would work fine (a variation of this plan is used in ultrasound scans).
There are other ways of doing this too...