A great question Mahonix! Let me try to explain that. Technically, an API gateway is a standalone component which would only provide features for exposing backend services through a single point by acting as a proxy server and extensions for applying security, throttling, statistics publishing, etc. In most cases an API gateway would need to be manually configured according to the backend services which need to be exposed similar to configuring a load balancer.
In contrast, as I have explained in this arcticle an API Management system would include an API gateway and a collection of other components for providing management capabilities for APIs out of box as services. This may include API security features, subscription management, circuit breaking, throttling, analytics, tracing, anomaly detection, fraud detection, etc. Hope this would clarify your question. Let me know if you need any further clarifications. Thanks!