Azure provides built-in diagnostics mechanism (App Service Logs) to assist with debugging an App Service app. There basically you will have to go through the logs to see what is happening.
Application Insights as the name suggests (Insights) provides a lot more features on top of just being able to view logs. Most importantly you can query the logs. Have a look at App Insights Overview.
To answer your question, I use App Service Logs when an App Service is failing to start, etc. For everything else, I use Application Insights.