If you go to Azure webapp, and on the left hand panel select Application Insights. Then View Application Insights Data and then click the Availability on the left hand panel, you can add new tests. Basically, here you can specify the health/ping endpoint for the site. You can also here configure some associated rules for the alerts.
Now, Azure has got a new functionality which is called Health Check on the webapp. All you have to do is enable it, and give it your health/ping endpoint. Then you can also configure rules here.
With both methods, the health endpoint is triggered by azure and if something is not right based on the alert rules you get an alert message.
But what is the difference between the two approaches?