What is the difference between Dielectric and Capacitive Polarizable Continuum Model?
Short answer: It matters most that you are using a modern solvation model than which one

Slightly longer answer: If you have the time, try a few solvation models, compare with experiment and see which gives you the best results. In my experience, modern models do fairly well for TDDFT calculations. Be careful about the atomic radii used and whether the computational program generates a reasonable solvation surface.

Also keep in mind that not every program supports both D-PCM and C-PCM. Many packages just implement one solvation model.

There are a number of excellent reviews on polarizable continuum solvation models. As you're probably aware, they are a really efficient way to handle molecular solvation in calculations, since the details of the solvent are largely ignored. Instead, the solvent is treated as a dielectric environment, and you handle polarization between the molecule and the solvent self-consistently. That is, the solute molecule is treated at normal atomistic detail, creates a solvation shell (usually treated by a set of overlapping spheres) and the electrostatics of the molecule influences the solvent, and vice-versa.

Now, I've never seen "C-PCM" as a "capacitive" PCM. Usually people refer to the "C" as a conductor-style model. That is, in a C-PCM model, the dielectric constant is set to $\infty$ but then you re-scale the charges on the molecule later.

There are more than just these techniques. The main problem is that it's very hard to bench-mark solvation energies.

Some great recent reviews:


"Thirty years of continuum solvation chemistry: a review, and prospects for the near future." Theoretical Chemistry Accounts (2004) v. 112(4) pp. 184-203
"Quantum Mechanical Continuum Solvation Models." Chem. Rev., 2005, 105(8), pp 2999–3094
"Perspective on Foundations of Solvation Modeling: The Electrostatic Contribution to the Free Energy of Solvation" J. Chem. Theory Comput., 2008, 4 (6), pp 877–887
"A Universal Approach to Solvation Modeling" Acc. Chem. Res., 2008, 41 (6), pp 760–768
"The COSMO and COSMO-RS solvation models" WIREs Comput Mol Sci, (2011) 1: 699–709
"Polarizable continuum models." WIREs Comput Mol Sci (2012) 2: 386–404.