I have been working with Enki Education in our home for 7 years. In my initial exploration of schooling I looked at Waldorf schools. I stumbled across Enki online and found it was a great fit for our family. My answers here come from a limited experience with Waldorf and a deep experience working with the Enki approach.
These are the differences that stood out for me:
Enki does not have a foundation world view of anthroposophy.
Enki is a global curriculum in which all cultures and world views have in depth study.
Enki uses discovery learning in all content.
In Enki movement work is informed by the field of Sensory Ingration (among other things as well) not eurythmy.
The similarities are that both have a foundation/world view that is well defined and all that is brought to the child is informed by this view. They are arts rich and give attention to environment, and rhythms.