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The transition between necking-mediated tensile failure of glasses, at elevated temperatures and/or low strain-rates, and shear-banding-mediated tensile failure, at low temperatures and/or high strain-rates, is investigated using tensile experiments on metallic glasses and atomistic simulations. We experimentally and simulationally show that this transition occurs through a sequence of macroscopic failure patterns, parametrized by the ultimate tensile strength. Quantitatively analyzing the spatiotemporal dynamics preceding failure, using large scale atomistic simulations corroborated by experimental fractography, reveals how the collective evolution and mutual interaction of shear-driven plasticity and dilation-driven void formation (cavitation) control the various macroscopic failure modes. In particular, we find that at global failure, the size of the largest cavity in the loading direction exhibits a nonmonotonic dependence on the temperature at a fixed strain rate, which is rationalized in terms of the interplay between shear- and dilation-driven plasticity. We also find that the size of the largest cavity scales with the cross-sectional area of the undeformed sample. These results shed light on tensile failure of glasses, and highlight the need to develop elasto-plastic constitutive models of glasses incorporating both shear- and dilation-driven irreversible processes.