Selfhosted

Some additional benefits also are the management of secrets. In compose you will shove them inside a .ENV file if not directly inside the compose file, while in Kubernetes you can use the secrets resource or even plug in Vault relatively easily. Stateful storage is also better handled. Named volumes are nasty to keep track of, backup and it's not possible to spread them across multiple devices (as in disks) while bind mounts are insecure in general. Kubernetes provides a storage abstraction which is easier to manage.

Obviously the big advantage comes when you want to run stuff on multiple devices to spread the load (or because the one box is saturated), since with compose you would need completely custom and independent setups.

Finally, I would say that running compose makes it much harder to have a monitoring stack supporting your services, since you will need to do all the plumbing for metrics endpoints yourself. And - very last - you can have admission controllers in Kubernetes that prevent certain configuration (e.g. Kyverno with a bunch of default policies), while with compose you need to manually vet every compose file and image (for example, to ensure it doesn't run as root).

That said, compose is perfect to get started and to run stuff on one machine.