As with many words in the English language, you have to consider which definition applies. In this context, "command" is fairly straightforward:
command (v): Give an authoritative or peremptory order (instruction)
"Order" has many more possible meanings:
1. Give an authoritative instruction to do something
2. Request (something) to be made, supplied, or served
When talking about authoritative instructions, "command" and "order" are synonymous.
I command/order you to sit down.
This doesn't apply only to the military. Anyone can "order" someone else within their area of authority, such as a guard to a prisoner, or a police officer to a citizen. In some schools it would be acceptable for a teacher or administrator to "order" a student to do something, or an upperclassman can "give orders" to younger students. Some organizations or religious groups allow those of higher rank to "command" those of lower rank. And so on.
Note the importance of "area of authority". Anyone can order anyone else to do something, but without authority to back it up, there's no reason why that order would be obeyed. For example, it wouldn't make sense for someone of lower rank in the military to order someone of higher rank, unless the relationship was complicated in some way that would be explained in context.
Meanwhile, there is a significant difference in meaning between "give an order" and "place an order". "Placing an order" (with a business or service) amounts to a request, not a demand.
You were late so I ordered us some pirogis.
We're almost out of paper towels so I ordered some more from Amazon.
Call up the printer and order us another thousand copies of our brochure.
and so on. The business that "takes the order" doesn't have to fulfill it, but since normally that's how they get paid, they are personally motivated to do so.
You may find it interesting to reference an etymology dictionary, which can help you understand why certain English words seem to have multiple meanings. For example, the original meaning of "order" was strictly "command", and its use to mean "formally request" is fairly recent (from 1837).