Both swarm in spring and both leave sawdust. The difference is visible in ten seconds if you know where to look.
Termite swarmers have four wings of equal length, a straight waist and short, straight antennae. Carpenter ant swarmers have a pinched waist, elbowed antennae, and front wings noticeably longer than the back pair. Shed wings on a windowsill in April are the single most common thing we get sent photos of, and the identification is almost always possible from that alone.
The reason it matters is cost. Carpenter ants nest in wood that is already damp and damaged — you fix the moisture, treat the nest, and you are done. Subterranean termites are a structural problem that needs a barrier or a bait system and an inspection schedule. Getting the species wrong means paying for the wrong treatment twice.