Ubuntu Mini Remix vs Ubuntu Minimal CD

Taken from https://help.ubuntu.com/community/Installation/MinimalCD
The Minimal CD will download packages from online archives at installation time instead of providing them on the install CD itself.

This is the main difference with Ubuntu Mini Remix, which is a self contained distro with the basic Ubuntu packages (list here https://answers.launchpad.net/ubuntu-mini-remix/+faq/33) and does not depend on anything else, providing all the basic ubuntu packages to be able to create your own distro being sure you've all you need.