We make a comparison study of the linear and nonlinear diffraction by a periodically poled lithium niobate (PPLN) dual linear-nonlinear thin-plate grating with weak surface corrugation against four types of white light sources. These are the ordinary halogen lamp thermal radiation source (WL1), silica photonic crystal fiber supercontinuum nanosecond laser source (WL2), and two femtosecond white lasers made by a Ti:sapphire pump laser beam passing through a fused silica plate with significant spectrum broadening (WL3) and through a cascaded silica plate and chirped PPLN crystal with ultrabroadband second-harmonic generation (WL4). The experiments show that the coherence, the peak power, the spectral bandwidth, profile and flatness of pump white light all contribute to shape the characteristics of linear and nonlinear optical diffraction pattern. In particular, for WL1, WL2, and WL4, the peak power is not sufficiently large, thus only linear diffraction occurs. For incoherent WL1, the interference is absent. Remarkably, when femtosecond white laser with sufficiently large peak power (WL3) shines upon the PPLN grating, bright and sharp linear Bragg scattering spots, dispersive colored linear diffraction patterns, and nonlinear Cherenkov radiation nonlinear diffraction patterns are observed simultaneously by naked eyes. The experiments would enrich the basic physical and optical understanding of linear and nonlinear optical diffraction characteristics of ultrabroadband white laser sources.