TL;DR: projection describes how strong a perfume is perceived by people close to you; sillage refers to the scent trail that lingers in the air after you've passed. A perfume can have strong projection and weak sillage, or the reverse — and neither is the same as longevity (how long it lasts).
Projection: what people who approach you can smell
Projection describes the distance at which a perfume is noticeable around you: a person has to be present and close enough to smell a perfume with strong projection. This is sometimes called the scent "bubble" a perfume gives off.
Sillage: the trail left behind after you've passed
The word comes from French and literally means the wake left by a ship. In perfumery, sillage is the scent trail a person perceives after you've already moved past them — without needing to be there at the moment the scent is smelled. A perfume with strong sillage "lingers" in a room or hallway long after the wearer has left.
Why these two concepts are so often confused
Projection and sillage are related but distinct: a perfume can project strongly up close without leaving a lasting trail once the person has moved away. Conversely, some perfumes that are more discreet up close still leave a surprisingly noticeable sillage behind them.
And what about longevity?
Longevity (or wear time) is yet another concept: it measures how long a perfume stays perceptible on your skin, independent of its projection or sillage. A perfume can easily last eight hours on skin while having discreet projection — these three characteristics (projection, sillage, longevity) vary independently of each other depending on the perfume's composition.
Why it's useful to know these distinctions
Understanding these concepts helps you choose a perfume better suited to the context: strong projection or noticeable sillage can be perfect for a night out, but become a liability in an enclosed workspace where a more contained perfume is generally preferable.