The relative stability of alternative carbon nanoparticles with icosahedral symmetry, such as diamond-like nanocrystallites and multiwalled fullerenes (onions), is investigated using the parametric model and the density functional tight-binding (DFTB) method. It is demonstrated that an increase in the size of particles and their hydrogenation favor the stabilization of diamond-like nanocrystallites. The formation of “intermediate” nanostructures consisting of diamond-like nanocrystallites inside the fullerene cage is revealed. The electronic spectra of icosahedral carbon nanoparticles are calculated.