We present maps and radial profiles of the gas temperature in the nearby galaxy clusters A2199 and A496, which have the most accurate ASCA spectral data for all hot clusters. These clusters are relaxed and can provide reliable X-ray mass measurements under the assumption of hydrostatic equilibrium. The cluster average temperatures corrected for the presence of cooling flows are 4.8+-0.2 keV and 4.7+-0.2 keV (90% errors), respectively. Outside the central cooling flow regions, the radial temperature profiles are similar to those of the majority of nearby relaxed clusters. They are accurately described by polytropic models with gamma=1.17+-0.07 for A2199 and gamma=1.24+-0.09 for A496. We use these polytropic models to derive accurate total mass profiles. Within r=0.5/h Mpc, which corresponds to a radius of overdensity 1000, the total mass values are 1.45+-0.15 10^14 /h Msun and 1.55+-0.15 10^14 /h Msun. These values are 10% lower than those obtained assuming constant temperature. The values inside a gas core radius (0.07-0.13/h Mpc) are a factor of >1.5 higher than the isothermal values. The gas mass fraction increases with radius (by a factor of 3 between the X-ray core radius and r_1000) and at r_1000 reaches values of 0.057+-0.005 and 0.056+-0.006 h^-3/2 for the two clusters, respectively. Our mass profiles within r_1000 are remarkably well approximated by the NFW universal profile. Since A2199 and A496 are typical relaxed clusters, the above findings should be relevant for most such systems. In particular, the similarity of the temperature profiles in nearby clusters appears to reflect the underlying universal dark matter profile. The upward revision of mass at small radii will resolve most of the discrepancy between the X-ray and strong lensing mass estimates. (Abridged)
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