  We analyze the Near Infrared ($\sim0.8-1\mu$m) rest-frame morphologies of
galaxies with $\log M_*/M_\odot>9$ in the redshift range $0<z<6$, compare with
previous HST-based results and release the first JWST-based morphological
catalog of $\sim20,000$ galaxies in the CEERS survey. Galaxies are classified
into four main broad classes -- spheroid, disk+spheroid, disk, and disturbed --
based on imaging with four filters -- $F150W$, $F200W$, $F356W$, and $F444W$ --
using Convolutional Neural Networks trained on HST/WFC3 labeled images and
domain-adapted to JWST/NIRCam. We find that $\sim90\%$ and $\sim75\%$ of
galaxies at $z<3$ have the same early/late and regular/irregular
classification, respectively, in JWST and HST imaging when considering similar
wavelengths. For small (large) and faint objects, JWST-based classifications
tend to systematically present less bulge-dominated systems (peculiar galaxies)
than HST-based ones, but the impact on the reported evolution of morphological
fractions is less than $\sim10\%$. Using JWST-based morphologies at the same
rest-frame wavelength ($\sim0.8-1\mu$m), we confirm an increase in peculiar
galaxies and a decrease in bulge-dominated galaxies with redshift, as reported
in previous HST-based works, suggesting that the stellar mass distribution, in
addition to light distribution, is more disturbed in the early universe.
However, we find that undisturbed disk-like systems already dominate the
high-mass end of the late-type galaxy population ($\log M_*/M_\odot>10.5$) at
$z\sim5$, and bulge-dominated galaxies also exist at these early epochs,
confirming a rich and evolved morphological diversity of galaxies $\sim1$ Gyr
after the Big Bang. Finally, we find that the morphology-quenching relation is
already in place for massive galaxies at $z>3$, with massive quiescent galaxies
($\log M_*/M_\odot>10.5$) being predominantly bulge-dominated.
