anonymous user Welcome to Microsoft Q & A Community Forum.
Currently App Service Health Check logs are available only in Azure Monitor. If you are using workspace-based Application insights, kindly note that with workspace-based Application Insights resources data is stored in a Log Analytics workspace with other monitoring data and application data. This simplifies your configuration by allowing you to more easily analyze data across multiple solutions and to leverage the capabilities of workspaces.
As per my understanding, both the features checks the availability of the web application hosted on Azure App service. While health check ensures to remove unhealthy instances (only if instance count >1) and alert where as Web tests checks the responsiveness of the site from selected locations in test.