We study families of open chaotic maps that classically share the same asymptotic properties-forward and backward trapped sets, repeller dimensions, and escape rate-but differ in their short time behavior. When these maps are quantized we find that the fine details of the distribution of resonances and the corresponding eigenfunctions are sensitive to the initial shape and size of the openings. We study phase space localization of the resonances with respect to the repeller and find strong delocalization effects when the area of the openings is smaller than h.