In this work, we investigate the inductive biases that arise from learning multiple tasks, either simultaneously (multi-task learning, MTL) or sequentially (pretraining and subsequent finetuning, PT+FT). We describe novel implicit regularization penalties associated with MTL and PT+FT in diagonal linear networks and single-hidden-layer ReLU networks. These penalties indicate that MTL and PT+FT induce the network to reuse features in different ways. 1) Both MTL and PT+FT exhibit biases towards feature reuse between tasks, and towards sparsity in the set of learned features. We show a "conservation law" that implies a direct tradeoff between these two biases. Our results also imply that during finetuning, networks operate in a hybrid of the kernel (or "lazy") regime and the feature-learning ("rich") regime identified in prior work. 2) PT+FT exhibits a novel "nested feature selection" behavior not described by either the lazy or rich regimes, which biases it to extract a sparse subset of the features learned during pretraining. This regime is much narrower for MTL. 3) PT+FT (but not MTL) in ReLU networks benefits from features that are correlated between the auxiliary and main task. We confirm our insights empirically with teacher-student models. Finally, we validate our theory in deep neural networks trained on image classification tasks, finding that they may exhibit a nested feature selection regime. We also introduce a practical technique -- weight rescaling following pretraining -- and provide evidence that this method can improve finetuning performance by inducing the network to operate in the nested feature selection regime.