Temporal reasoning is the task of predicting temporal relations of event pairs with corresponding contexts. While some temporal reasoning models perform reasonably well on in-domain benchmarks, we have little idea of the 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 if systems can correctly understand the effect of incremental changes. Specifically, TODAY makes slight context changes for given event pairs, and systems need to tell how this subtle contextual change will affect temporal relation distributions. To facilitate learning, TODAY also annotates human explanations. We show that existing models, including GPT-3, 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 and encourage models to use more appropriate signals during training and outperform across several benchmarks. TODAY can also be used to train models to solicit incidental supervision from noisy sources such as GPT-3 and moves farther towards generic temporal reasoning systems.