How is nuclear fusion in a main-sequence star different from nuclear fusion in a giant star?

The difference is the stage and phase at which they’re in.
Because Giant stars themselves when compared to Main Sequence Stars the stage at which they are in differs from that of a main sequence star because giant stars themselves are now in the stage and phase in the life cycle of a star wherein the Thermonuclear Fusion within the star itself fuses Helium together to create an even heavier element which is Carbon because there is no more Hydrogen left to fuse them together in order to create Helium (CNO Cycle) and as a result the Thermonuclear Fusion within/of the star itself then begins to fuse Helium together which in turn creates Carbon.
Main Sequence stars themselves are in the stage in the life cycle of a star wherein there is still enough Hydrogen to be fused together which then in turn creates Helium (Which is the Proton-Proton chain) one example of a Main Sequence star is our sun because our sun still has enough Hydrogen left to fuse them together in order to create Helium and the process itself can last for as long as there is still enough Hydrogen left in order to fuse them together just to create Helium.