When choosing between SQL Server Express and PostgreSQL for .NET Core 2 web applications, consider factors like integration, scalability, and platform support. SQL Server Express offers seamless integration with Microsoft tools, making it ideal for small-to-medium projects in Windows environments. However, its limitations in features and scalability may pose challenges for larger applications. On the other hand, PostgreSQL provides a cross-platform solution with advanced features, better concurrency management, and no size limitations, making it a strong choice for growing applications. The decision should depend on your project's needs, performance requirements, and your team's familiarity with the databases.