The JEE standard forbids the direct management of connections in JEE applications. But if that's not reason enough to delegate connection management to the servlet container, consider that the container is both better positioned and better equipped to manage resources not just for a single class, but across classes in an application. Further, mis-managing connections in a JEE class can lead to connection leaks which could compound into a denial of service.
This rule raises an issue for each use of a DriverManager in a servlet class.