There seems to be a current trend in java space to move away from deploying java web applications to a java servlet container (or application server) in the form of a war file (
An interesting question. This is just my view on the topic, so take everything with a grain of salt. I have occasionally deployed and managed applications using both servlet containers and embedded servers. I'm sure there are still many good reasons for using servlet containers but I will try to just focus on why they are less popular today.
Short version: Servlet containers are great to manage multiple applications on a single host but don't seem very useful to manage just one single application. With cloud environments, a single application per virtual machine seems preferable and more common. Modern frameworks want to be cloud compatible, therefore the shift to embedded servers.
So I think cloud services are the main reason for abandoning servlet containers. Just like servlet containers let you manage applications, cloud services let you manage virtual machines, instances, data storage and much more. This sounds more complicated, but with cloud environments, there has been a shift to single app machines. This means you can often treat the whole machine like it is the application. Each application runs on a machine with appropriate size. Cloud instances can pop up and vanish at any time which is great for scaling. If an application needs more resources, you create more instances.
Dedicated servers on the other hand usually are powerful but with a fixed size, so you run multiple applications on a single machine to maximize the use of resources. Managing dozens of application - each with their own configurations, web servers, routes and connections etc. - is not fun, so using a servlet container helps you to keep everything manageable and yourself sane. It is harder to scale though. Servlet containers in the cloud don't seem very useful. They would have to be set up for each tiny instance, without providing much value since they only manage a single application.
Also, clouds are cool and non-cloud stuff is boring (if we still believe the hype). Many frameworks try to be scalable by default, so that they can easily be deployed to the clouds. Embedded servers are fast to deploy and run so they seem like a reasonable solution. Servlet containers are usually still supported but require a more complicated set up.
Some other points: