Here's three example use cases to hopefully show the differences:
Application Gateway: I have a web server/ web farm in a single Azure Region and want to ensure availability, balance traffic coming in, and maybe take advantage of features such as SSL offload and Web Application Firewall to secure the website.
Load Balancer: I have several application or database servers spread across Azure availability Zones (within a region) and I want to balance traffic coming in and provide high availability.
Traffic Manager: I want to build a globally resilient service allowing for a total region failure.
You can also use these solutions together, and with Azure Front Door.
https://docs.microsoft.com/en-us/azure/architecture/guide/technology-choices/load-balancing-overview... has a good walkthrough and a flowchart to show how to choose a load balancing solution.