Resonance-enhanced four-wave mixing and generation of sum-frequency field in xenon have been studied under two-color excitation by spatially coherent and incoherent conical laser beams. Comparisons of analogous results with the two-color excitation in ordinary geometry of focussed beams and in one-color Bessel beams are made. It has been shown that with incoherent laser beams the four-wave mixing process is much less degraded in conical excitation geometry where an efficient generation of sum-frequency field can be obtained despite of multiple wave-front aberrations. Such a tolerance of conical beams results from their specific structure of coherent focal domains which are extended significantly along the beam axis. In two-color conical beams the four-wave mixing process is driven within the central lobe of the coherent Bessel beam, where a few of long coherent domains from the incoherent beam are embedded. It preserves a relatively high degree of coherence for excitation processes. Numerical simulation of sum-frequency excitation profiles for one-and two-color conical beams has been carried out and good agreement with experimental observations has been obtained.