Which NetApp product should be recommended for monitoring and optimizing resources in both private and public cloud environments?

The most appropriate choice for monitoring and optimizing resources in both private and public cloud environments is Active IQ Unified Manager. This product is specifically designed to provide comprehensive insights into the health and performance of storage systems. It enables organizations to proactively manage their infrastructures by monitoring performance metrics, offering recommendations for optimization, and helping in capacity planning.

Active IQ Unified Manager integrates with both on-premises and cloud-based systems, making it versatile for hybrid cloud environments. This capability allows users to gain visibility into resource usage across different storage solutions and ensure that performance aligns with business objectives.

In comparison, while Cloud Insights is also aimed at monitoring and provides visibility into cloud infrastructures, it focuses more on operational insights rather than detailed management. Cloud Manager is geared more towards managing cloud-based workflows and configurations rather than direct performance monitoring or optimization. The Cloud Sync service primarily facilitates data synchronization between on-premises and cloud, which does not directly pertain to monitoring or optimization of resources.

Thus, Active IQ Unified Manager's features of in-depth monitoring, predictive analytics, and optimization recommendations make it the best fit for the given requirements in cloud environments.