Web search engines have long been the primary gateway to online information, but generative AI chatbots are emerging as alternative tools. Yet we lack empirical understanding of how users choose between these tools and how their information-seeking behaviors differ across these contexts. Understanding how users select, use, and assess information retrieval (IR) systems is essential for evaluating them in ways that reflect user expectations across contexts. To shed light on user perceptions and usage of IR systems, we surveyed 84 participants about their recent search engine and GenAI chatbot use, analyzing task types, contextual triggers, outcomes, and tool preferences. Our findings reveal that users approach search engines and GenAI chatbots with distinct philosophies. With search engines, they follow a 'Browse and Verify' approach, prioritizing reliability and source verifiability. With GenAI chatbots, they adopt a 'Delegate and Consume' approach, valuing quick summarization and personalized content. Despite these stated preferences, actual usage shows that users frequently turn to GenAI chatbots for actionable guidance in high-stakes domains such as health and relationship matters, hinting at a divergence between stated philosophies and real-world behavior. These findings underscore the need for IR systems that support reliable information assessment, transparent sources, and user-guided evaluation.