Why doesn't diamond conduct electricity but silicon does?
I'm not a solid-state physicist. Yet, this is how I understand this problem, so bare with me.

According to this article:


  Every solid has its own characteristic energy-band structure. This variation in band structure is responsible for the wide range of electrical characteristics observed in various materials. In semiconductors and insulators, electrons are confined to a number of bands of energy, and forbidden from other regions. The term "band gap" refers to the energy difference between the top of the valence band and the bottom of the conduction band. Electrons are able to jump from one band to another. However, in order for an electron to jump from a valence band to a conduction band, it requires a specific minimum amount of energy for the transition. The required energy differs with different materials. Electrons can gain enough energy to jump to the conduction band by absorbing either a phonon (heat) or a photon (light).


This forbidden gap plays a major role in determining the electrical conductivity of material. The materials can be classified into three types based on the forbidden gap. they are:


Insulators: The materials, which does not allow the flow of electric current through them are called insulators. The band gap between the valence band and conduction band is very large in insulators, which is approximately equal to $\pu{15 eV}$. Normally, in insulators the valence band is fully occupied with electrons due to sharing of outer most orbit electrons with the neighboring atoms whereas its conduction band is empty, i.e., no electrons are present in the conduction band.
Semiconductors: The material, which has shown electrical conductivity between that of a conductor and an insulator is called a semiconductor. The band gap of semiconductors is is very small, and it falls between that of valence band and conduction band. The value of this forbidden gap is about $\pu{1-3 eV}$. As a rule of thumb, if the material has $\Delta E_\mathrm{gap} \le \pu{3.0 eV}$, it is considered as a semiconductor (Semiconductors- Band Gaps, Colors, Conductivity and Doping). 
Conductors: In a conductor, the valence and conduction bands are overlapped each other. Therefore, there is no forbidden gap in a conductor (the value is generally negative). Thus, the valence band electrons to move into conduction band needs only a little amount of applied external energy.


Experimental values of $\Delta E_\mathrm{gap}$ for diamond and silicon are $\pu{5.48 eV}$ and $\pu{1.17 eV}$, respectively (Ref.1). Keep in mind that band gap get narrowed when going down on a group in periodic table. For example, $\Delta E_\mathrm{gap}$ of germanium is $\pu{0.74 eV}$ (Ref.1).

Therefore, silicon and germanium behave like semiconductors, while diamonds fall into insulator category in general (yet, please refer to  Ed V's comment above for better definition).

References:


Fabien Tran, Peter Blaha, "Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential," Phys. Rev. Lett. 2009, 102, 226401, 4 pages (DOI: 10.1103/PhysRevLett.102.226401).