What is the difference between docker-compose up and docker-compose start?

docker-compose start
(https://docs.docker.com/compose/reference/start/)

Starts existing containers for a service.

docker-compose up
(https://docs.docker.com/compose/reference/up/)

Builds, (re)creates, starts, and attaches to containers for a service.

Unless they are already running, this command also starts any linked services.

The docker-compose up command aggregates the output of each container
(essentially running docker-compose logs -f). When the command exits,
all containers are stopped. Running docker-compose up -d starts the
containers in the background and leaves them running.

If there are existing containers for a service, and the service’s
configuration or image was changed after the container’s creation,
docker-compose up picks up the changes by stopping and recreating the
containers (preserving mounted volumes). To prevent Compose from
picking up changes, use the --no-recreate flag.

For the complete CLI reference:
https://docs.docker.com/compose/reference/