Differences between API Gateways and ESBs?

The company I work for is evaluating some middleware solutions for governace, metering and security of web services. Currently, we're using an Enterprise Service Bus (ESB) for this purpose, but some cool guys in management decided they are going to deploy some API Management Middleware.

I researched a bit about these API Management (aka API Gateway) Solutions but couldn't find the difference between them and actual ESBs. I evaluated some white papers from Mule, WSO2, Oracle etc, but the features offered by both products seem to be almost the same. The question is, what an API Management can do that an ESB cannot do and vice-versa? What value can be added to an IT Infrastructure by replacing an ESB for an API Gateway?