  In a previous paper, we assembled a collection of medium-resolution spectra
of 35 carbon stars, covering optical and near-infrared wavelengths from 400 to
2400 nm. The sample includes stars from the Milky Way and the Magellanic
Clouds, with a variety of $(J-K_s)$ colors and pulsation properties. In the
present paper, we compare these observations to a new set of high-resolution
synthetic spectra, based on hydrostatic model atmospheres. We find that the
broad-band colors and the molecular-band strengths measured by
spectrophotometric indices match those of the models when $(J-K_s)$ is bluer
than about 1.6, while the redder stars require either additional reddening or
dust emission or both. Using a grid of models to fit the full observed spectra,
we estimate the most likely atmospheric parameters $T_\mathrm{eff}$, $\log(g)$,
$[\mathrm{Fe/H}]$ and C/O. These parameters derived independently in the
optical and near-infrared are generally consistent when $(J-K_s)<1.6$. The
temperatures found based on either wavelength range are typically within
$\pm$100K of each other, and $\log(g)$ and $[\mathrm{Fe/H}]$ are consistent
with the values expected for this sample. The reddest stars ($(J-K_s)$ $>$ 1.6)
are divided into two families, characterized by the presence or absence of an
absorption feature at 1.53\,$\mu$m, generally associated with HCN and
C$_2$H$_2$. Stars from the first family begin to be more affected by
circumstellar extinction. The parameters found using optical or near-infrared
wavelengths are still compatible with each other, but the error bars become
larger. In stars showing the 1.53\,$\mu$m feature, which are all
large-amplitude variables, the effects of pulsation are strong and the spectra
are poorly matched with hydrostatic models. For these, atmospheric parameters
could not be derived reliably, and dynamical models are needed for proper
interpretation.
