What are the connotations of production and manufacturing? In what situation would you prefer one over the other?
The difference is in how the item in question is produced.
To make or process (a raw material) into a finished product, especially by means of a large-scale industrial operation.
To make or process (a product), especially with the use of industrial machines.
To create, produce, or turn out in a mechanical manner: "His books seem to have been manufactured rather than composed" (Dwight Macdonald).
To concoct or invent; fabricate: manufacture an excuse.
See how industrial machines or mechanical way are explicitly mentioned in every meaning
Production is more general.
The act or process of producing: timber used for the production of lumber and paper.
The fact or process of being produced: a movie going into production.
The creation of value or wealth by producing goods and services.
The total output, as of a commodity: increased production at the plant.
Something produced; a product: "Of all the productions of land, milk is perhaps the most perishable" (Adam Smith).
A work of art or literature.
A work produced for the stage, screen, television, or radio.
A staging or presentation of a theatrical work: a new Broadway production of a musical.
A situation or display that is exaggerated or unduly complicated: made a production out of the birthday party.