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.