Before talking about load balancing, we need to be aware of a conceptual model named OSI model. The Open Systems Interconnection (OSI) model is a conceptual model that characterizes and standardizes the communication functions of a telecommunication or computing system. I am not getting too deep into it as I need only a part of the OSI model to explain Azure Load Balancer solutions. You need to know about the layers in OSI. In the below image, you need to count the layers from the bottom to the top; i.e. 4th layer is Transport and 7th layer is Application. Also notice the protocols under each layer, as the layer 4 has the network-related protocols like TCP, UDP, etc... and layer 7 has protocols like HTTP, SMTP, RIP, etc... Now, let us go through each of the Azure load balancing solutions to differentiate them.
Azure Load Balancer
This load balancer works on layer 4 of the OSI model. It works inside a single region. It has no access to the Application or Session details as they are in different layers under the OSI model. So, any load balancing features related to Application or Session are not available with it. Here, load balancing rules work at the port level as it knows only TCP/UPD, etc... of layer 4.
Azure Application Gateway
This load balancer works on layer 7 of the OSI model. Layer 7 is the Application layer and so has access to a lot more information compared to layer 4. So, Application Gateway can contain more granular load balancing rules based on URL, Domain, etc. and it is smarter load balancer than the Azure Load Balancer. It supports much more feature-packed load balancings like sticky sessions, connection affinity, etc. This too works under a single region.
Azure Traffic Manager
This load balancer works across regions. Azure Traffic Manager uses DNS to redirect requests to an appropriate geographical location endpoint. It works on the DNS level and distributes the load over multiple regions and data centers using the rules configured at the DNS level.
As per image below, Traffic Manager finds the nearest region and sends that back to the client via a DNS response.
Hope this article will help you to take decisions on Azure load balancing solutions based on how and where each of these Azure load balancing solutions work.