Usually, a server throwing an error would mean that it is aware there was an unexpected state, and is itself consciously not fulfilling the request by returning a 500 error, for example. It remains available to handle the next incoming request.

A server crashing implies that the server program or process itself has terminated, and is not able to handle further requests. This usually manifests as a 503 error by an upstream proxy server (nginx/apache/CDN/etc.).