"Set-in" vs "Flat" sleeve sewing -- is there really a difference?
To be absolutely honest, the difference is marginal in most (but not all) cases.
You probably won't even notice any difference in:

short sleeves
wide or baggy garments (like a tunic or pirate blouse)
garments where the shoulder seam extends far down the shoulder (like a kimono or most casual menswear)
garments sewn from stretchy fabric

Puffy sleeves that are gathered at the top of the shoulder should always be set-in sleeves because that allows you to place the gathers where they need to be. In theory you could gather the (open) sleeve first and still use the flat method, but I honestly find it easier to use the set-in method in this case.
In tight fitting garments that don't stretch (like formal business shirts or coats) a flat-sewn sleeve might be too tight to accommodate the shoulder. The shoulder seam usually sits on top of the shoulder and the sleeve points slightly down at an angle. But the human body doesn't have angles like that, there is still the round of the shoulder that has to fit in there:

The picture is a tiny bit exaggerated to make the difference visible. (Original image source)
The blue line represents the flat-sewn sleeve. The fabric forms a straight tube without any ease, but the round top of the shoulder somehow has to fit in there. Bulky people will notice the difference more than skinny ones.
The green line represents the set-in sleeve. It was cut wider than the armscye on purpose, then was slightly gathered on top to shorten the seamline to the same length as the armscye, before it was set in. Since the sleeve cap is a curve, some parts of it are cut on the bias and can compensate the gathering without puckering. The result is not a straight tube, but one that has a bulge at the top to accommodate the shoulder.
Here are some tutorials for set-in sleeves that clearly show the too-wide sleeve being gathered into the armscye:

Tutorial: Setting In Sleeves
Set in Sleeve : Perfect way to attach sleeves