API gateways are applied on Layer 7 of OSI model or you can say to manage traffic coming from outside network ( sometimes also called north/south traffic ) , whereas Service Mesh is applied to Layer 4 of OSI model or to manager inter-services communications ( sometimes also called as east/west traffic). Some examples of API Gateway features are Reverse Proxy,Load Balancing , Authentication and Authorization , IP Listing , Rate-Limiting etc.
Service Mesh, on the other hand, works like a proxy or a side-car pattern which de-couples the communication responsibility of the service and handles other concerns such as Circuit breaker , timeouts , retries , service-discovery etc.
If you happen to use Kubernetes and Microservices then you might want to explore other solutions such as Ambassador + Istio Or Kong which works as Gateway as well as Service Mesh.