New Relic monitors your application code execution and your end user's experiences. New Relic Plugins monitor anything - other parts of your application stack that our "stock" agents don't monitor, parts of your business process, whatever.

Azure's Management Portal monitors your infrastructure. New Relic also does a little infrastructure monitoring.

Overall, they do different things - and, New Relic is both free to use and super-easy to install, so perhaps you can try it out and see what you think?