Databases and search engines (open web) serve different purposes. One isn't better than the other. The choice to use one or the other depends upon what information you need.
Databases - best for
- In-depth, subject specific research
- Published information (copyrighted, licensed, proprietary)
- Reliable and relevant information
- Accessibility of full-text
- Ability to filter and refine results
Search Engines - best for
- Finding information from/by governments, organizations, groups and individuals
- Finding statistical information (e.g., Census demographics)
- Trending/current topics and popular topics
- General starting point to learn about a topic
- Finding images
