Though API management definition is still evolving but API management is defined as transforming APIs to reach to your target audience [ here is a good description - http://searchcloudapplications.techtarget.com/definition/API-management].
This technology has it's root in SOA but different from ESB.
ESB is more for system-to-system integration but API tends to be securely exposing your internal systems in a managed fashion to wider audience - we call them "developers".
ESB tends to be equipped with many adapters and strong message oriented middleware for supporting different interaction patterns. It is also generally couple with business process management software to automate internal processes by integrating multiple services or systems.
API management also does some integrations and orchestrations but focus is more simplifying the interfaces to easier consumption of the services - that's why it is always come with developer on-boarding capabilities, security , caching , api design , oauth etc.
Apigee gateway service [ API management platform ] has support for limited JMS and SMTP functionality serving our diverse customer base and our technology stack is capable of building other protocol support.
Many API management products [ including Apigee gateway ] also include API analytics to help you getting insight of API program and API traffic.