The development of CLIP [Radford et al., 2021] has sparked a debate on whether adding language supervision can yield vision models with more transferable representations than traditional image-only methods. Our work studies this question through a carefully controlled comparison of two approaches, in terms of their ability to learn representations that generalize to downstream classification tasks. We find that when the pre-training data meets certain criteria---it is sufficiently large and contains descriptive captions with low variability----image-only methods do not match CLIP's performance even when they are trained with more image data. However, contrary to what one might expect, there are practical settings in which these criteria are not met, wherein added supervision through captions is actually detrimental.Motivated by our findings, we devise simple data and algorithmic interventions to improve the transfer performance of CLIP-style models.