Difference between By order of someone and Under the orders of someone

by order of someone—according to directions given by a proper authority:

• By order of the Minister a special operation of police is being carried out.

under the orders of someone—

1. (Military) commanded by a person; being smb.’s subordinate:

• At the battle of Waterloo he was under the orders of Marshal Grouchy.

2. = by order of someone:

• In some cases it may be possible to release the prisoner on probation, but this is permitted only under the orders of the Provincial Government.