These two words get used interchangeably every day in barbershops, and barbers just quietly correct for it. The confusion is understandable - both involve hair getting shorter toward the neckline and ears - but they describe two different things, and booking for one when you want the other will leave you looking at a haircut that's close but not quite right.
A taper is the baseline. Every professional haircut has some degree of taper to it. It means the hair gradually shortens as it moves down toward the hairline, usually just around the ears and the back of the neck. A tapered cut still has noticeable length near the bottom - you're not seeing skin, just a clean transition from longer on top to shorter at the edges. It's conservative, classic, and grows out well. Most traditional cuts - the side part, a textured crop worn looser, a business cut - are tapered by nature.
A fade takes that graduation further. Where a taper stops before the skin, a fade blends all the way down to near-skin or full skin, usually a few inches up the sides. The transition is tighter, the contrast is higher, and the result is sharper. The fade line itself - where it sits on the head - is its own variable, which is why there are low fades, mid fades, and high fades, each changing the proportions of the whole cut.
Who suits what isn't strictly about face shape, though that plays a role. A taper is lower maintenance in terms of visual upkeep - it grows out more forgivingly and can go four to six weeks before looking rough. A fade starts showing regrowth faster because the blended skin or near-skin at the bottom gets fuzzy within two to three weeks. If you're someone who cuts every month no matter what, fades look sharp for longer proportionally. If you're stretching visits out, a taper might suit your schedule better.
To actually ask for what you want: tell your barber whether you want to see skin at any point. That one piece of information separates a taper from a fade more clearly than any terminology.