An API Gateway is a proxy provided for the client. The Gateway gives the client a consistent interface regardless of any changes within the internal system. It allows the internal system to change without affecting the client. The API Gateway can also provide consistent cross-cutting concerns such as security logging, reporting and API analytics.
An ESB (Enterprise Service Bus) provides a means for service-to-service communication. With this technique, services do not need to communicate with each other, reducing coupling. ESBs often use guaranteed messaging for inter-service communication.
Today, the Service Mesh pattern has become popular for microservices. A Service Mesh implementation can provide both an API Gateway and service-to-service communication, along with load balancing, security and many other features.
There are a lot of variations and implementation details, but this is the high-level difference.