The public DNS name always matches the public IP address.
The public IP address stays the same for an instance until it is terminated or stopped. A reboot does not change the public IP address.
If an EC2 instance is in a VPC, then it will retain the same public IP address across a stop and start.
If an EC2 instance that is not in a VPC is stopped and then started started again, it will probably receive a different public IP address.
Instances can fail. When you start a new instance to replace a failed or terminated instance, it will probably receive a different public IP address.
Because instances can fail, and because you may want to change the size of an instance (with a stop/start) it is not recommended to "ship an app that's wired to [the public IP address]" (or DNS name). Once your instance is stopped/terminated/failed another user could get that IP address assigned to their instance and all your traffic would go to them.
It is recommended to use Elastic IP Addresses to associate public services with your instance. You get to keep the Elastic IP address and you can assign it to any instance you want over time, even if it's the same instance after a stop/start.
Each Elastic IP Address comes with a public DNS name, but you would probably be better off mapping your own hostname to the Elastic IP address so that the name makes more sense to humans.
Here's a guide to Elastic IP Addresses:
Here's an article I wrote that talks about the differences between rebooting and stop/start of an instance:
Rebooting vs. Stop/Start of Amazon EC2 Instance
Here's an article I wrote that provides a reason you may want to stop/start an instance even though you don't think you will today:
Moving an EC2 Instance to a Larger Size