Conversational search offers an easier and faster alternative to conventional web search, while having downsides like lack of source verification. Research has examined performance disparities between these two systems in different settings. However, little work has considered the effects of variations within a given search task. We hypothesize that psychological distance - one's perceived closeness to a target event - affects information needs in search tasks, and investigate the corresponding effects on user preferences between web and conversational search systems. We find that with greater psychological distances, users perceive conversational search as more credible, useful, enjoyable, and easy to use, and demonstrate increased preference for this system. We reveal qualitative reasons for these differences and provide design implications for search system designers.