The simplicity Bias (SB) of neural networks, i.e.\ their tendency to represent simple functions, is a key factor in their generalization capabilities. Recent studies show that an excessive SB may harm performance on complex tasks, and the need for this bias varies across tasks. Many of these studies focus on simple models or synthetic tasks. It remains challenging to measure the SB in large models and little is known about the relevance of the SB to various image classification tasks. In this paper, we investigate the relationship between the SB in CLIP models and their performance across image classification tasks. First, we theoretically analyze the potential limitation of existing measures of complexity that have been used to characterize small models. To address this, we propose a frequency-aware measure capturing finer-grained SB differences. We validate this measure on CLIP models subjected to two recent SB-modulation methods, demonstrating that it is more informative and consistent than previous measures. Second, we examine the relation between the SB of those models and their performance across a range of image classification tasks, including zero-shot and fine-tuning settings. These experiments reveal a range of behaviors. For example, a stronger SB correlates with a better performance on OOD generalization than on adversarial robustness. These results highlight the benefits of aligning a model's inductive biases with the characteristics of the target task.