In the Star Trek: The Next Generation episode “The Pegasus” (S07E12), Pressman clarifies that he is in charge of the overall mission while Picard remains in command of the Enterprise.
What were the modifications of duties, responsibility, and authority as a consequence of that?
What did Pressman not take control of, and what did Picard lose control of?
Picard's usual responsibilities to maintain the safety of his ship and crew, to ensure that tasks onboard the Enterprise are performed by qualified personnel, and to obey the regulations of Starfleet remain unchanged. He's just got his boss close at hand, not a subspace message away.
First, some background: Typically, an admiral is in command of several ships, and has workspace (offices) on one particular ship (the flagship). In that circumstance the admiral conveys his orders to any of the ships in his command via those ships' chains of command, by ordering the captains or the person holding the conn in the captain's stead, as described in this article :
Just as we were about to sit down at the handsomely laid table, the ship started rolling steeply, so heavily that the table settings began sliding, even though the tablecloth had been dampened against just such a contingency.
It was a sizable formation, a dozen or more ships in all, and our course, homeward bound, was north, up our own Atlantic coast, when we came into the deep trough that set up the rolling. The admiral’s reaction was instantaneous. “This won’t do,” he said, and , turning to his aide, “Tell them to turn twenty degrees east.” The aide spoke into a telephone, giving an order that applied to the whole formation, flagship included.
The same article notes:
The admiral holds complete authority over all the ships in the unit, while the captain is absolute master of and responsible for his ship and all aboard her.
In summary, the Admiral defines the mission, via orders to the captain; the captain implements via orders to the crew.
When there's only one ship involved, things are a bit different, because the distinction between orders to the one vessel involved and orders to the crew of the vessel becomes a little blurry, but
In general, military personnel give orders only to those directly below them in the chain of command and receive orders only from those directly above them. A service member who has difficulty executing a duty or order and appeals for relief directly to an officer above his immediate commander in the chain of command is likely to be disciplined for not respecting the chain of command. Similarly, an officer is usually expected to give orders only to his or her direct subordinate(s), even if only to pass an order down to another service member lower in the chain of command than said subordinate.
Therefore, Pressman gives orders like "We need to enter the X system surreptitiously" or "I want information about that asteroid" and Picard chooses how to implement them, assigning crew to the roles involved. Pressman may give direct orders to the person holding the conn if Picard is off duty. But Pressman shouldn't give orders to other staff.
