Temporal reasoning is the task of predicting temporal relations of event pairs. While temporal reasoning models can perform reasonably well on in-domain benchmarks, we have little idea of these systems’ generalizability due to existing datasets’ limitations. In this work, we introduce a novel task named TODAY that bridges this gap with temporal differential analysis, which as the name suggests, evaluates whether systems can correctly understand the effect of incremental changes. Specifically, TODAY introduces slight contextual changes for given event pairs, and systems are asked to tell how this subtle contextual change would affect relevant temporal relation distributions. To facilitate learning, TODAY also annotates human explanations. We show that existing models, including GPT-3.5, drop to random guessing on TODAY, suggesting that they heavily rely on spurious information rather than proper reasoning for temporal predictions. On the other hand, we show that TODAY’s supervision style and explanation annotations can be used in joint learning, encouraging models to use more appropriate signals during training and thus outperform across several benchmarks. TODAY can also be used to train models to solicit incidental supervision from noisy sources such as GPT-3.5, thus moving us more toward the goal of generic temporal reasoning systems.