PostgreSQL is widely used in large systems where read and write speeds are crucial and data needs to validated. In addition, it supports a variety of performance optimizations that are available only in commercial solutions such as Geospatial data support, concurrency without read locks, and so on (e.g. Oracle, SQL Server).
Overall, PostgreSQL performance is utilized best in systems requiring execution of complex queries.
PostgreSQL performs well in OLTP/OLAP systems when read/write speeds are required and extensive data analysis is needed.
PostgreSQL also works well with Business Intelligence applications but is better suited for Data Warehousing and data analysis applications that require fast read/write speeds.
MySQL is a widely chosen for web based projects that need a database simply for straightforward data transactions. It is common, though, for MySQL to underperform when strained by a heavy loads or when attempting to complete complex queries.
MySQL performs well in OLAP/OLTP systems when only read speeds are required.
MySQL + InnoDB provides very good read/write speeds for OLTP scenarios. Overall, MySQL performs well with high concurrency scenarios.
MySQL is reliable and works well with Business Intelligence applications, as business intelligence applications are typically read-heavy.