Recent studies have shown that contrastive learning, like supervised learning, is highly vulnerable to backdoor attacks wherein malicious functions are injected into target models, only to be activated by specific triggers. However, thus far it remains under-explored how contrastive backdoor attacks fundamentally differ from their supervised counterparts, which impedes the development of effective defenses against the emerging threat. This work represents a solid step toward answering this critical question. Specifically, we define TRL, a unified framework that encompasses both supervised and contrastive backdoor attacks. Through the lens of TRL, we uncover that the two types of attacks operate through distinctive mechanisms: in supervised attacks, the learning of benign and backdoor tasks tends to occur independently, while in contrastive attacks, the two tasks are deeply intertwined both in their representations and throughout their learning processes. This distinction leads to the disparate learning dynamics and feature distributions of supervised and contrastive attacks. More importantly, we reveal that the specificities of contrastive backdoor attacks entail important implications from a defense perspective: existing defenses for supervised attacks are often inadequate and not easily retrofitted to contrastive attacks. We also explore several alternative defenses and discuss their potential challenges. Our findings highlight the need for defenses tailored to the specificities of contrastive backdoor attacks, pointing to promising directions for future research.