Elastic Beanstalk and Datadog are used by SolveBio to manage and monitor their applications on EC2, with a focus on Python and Node.js stacks. The company initially encountered issues integrating the two tools, but was able to resolve them by telling Elastic Beanstalk to stop and start the Datadog Agent during deployment. Once set up, the Datadog Agent can be deployed using environment configuration files and provides valuable insights into application performance, including metrics on AWS Elastic Beanstalk instances.