Interaction of relativistically intense axisymmetrically polarized (radially or azimuthally polarized) laser pulses (RIAPLP) with underdense plasma is shown experimentally and theoretically to be essentially different from the interaction of conventional Gaussian pulses. The difference is clearly observed in distinct spectra of the side-scattered laser light for the RIAPLP and Gaussian pulses, as well as in the appearance of a spatially localized strong side emission of second harmonic of the laser pulse in the case of RIAPLP. According to our analysis based on three-dimensional particle-in-cell simulations, this is a result of instability in the propagation of RIAPLP in uniform underdense plasma.