The James Webb Space Telescope's (JWST's) unprecedented combination of sensitivity, spatial resolution, and infrared coverage has enabled a new era of galaxy morphology exploration across most of cosmic history. We analyze the near-infrared (NIR $ rest-frame morphologies of galaxies with $ M_*/M_ in the redshift range of $0<z<6$, compare with previous HST-based results and release the first JWST-based morphological catalog of $ galaxies in the CEERS survey. We classified the galaxies in our sample into four main broad classes: spheroid, disk+spheroid, disk, and disturbed, based on imaging with four filters: $F150W$, $F200W$, $F356W$, and $F444W.$ We used convolutional neural networks (CNNs) trained on HST/WFC3 labeled images and domain-adapted to JWST/NIRCam. We find that $ and $ of galaxies at $z<3$ have the same early and late and regular and 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 $ Using JWST-based morphologies at the same rest-frame wavelength ($ 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 ($ M_*/M_ at $z and bulge-dominated galaxies also exist at these early epochs, confirming a rich and evolved morphological diversity of galaxies $ 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 ($ M_*/M_ being predominantly bulge-dominated.