application insights vs elastic (ELK)

I'm from App Insights team. I think the link provided by @rickvdbosch in a comment gives quite good perspective. It is 1+ years old at this point, so, some items regarding App Insights evolved since then.

I think App Insights and ELK are quite different offerings. The former is managed offering (you can set it up within couple minutes), focused on very broad range of out-of-the-box experiences (collecting incoming/outgoing requests, exceptions, smart alerts, availability monitoring, analytics, live metrics, application map, end-to-end transactions across apps).

My understanding of ELK is that it has very powerful UI visualization and powerful dashboards (though there are adapters for Kibana to work with Azure Monitor). For scenarios where there is a need to store a lot of data (highly loaded apps with adaptive sampling still store limited amount of data) ELK solution might be cheaper to run.

Final decision was to use ELK as servers already have all the configuration, because other team uses it and mainly because logging will need a lot customization.