Trace, vestige, track can all mean a visible or otherwise sensible sign left by something that has passed or has taken place.
Trace basically applies to a line (as of footprints) or a rut made by someone or something that has passed.
The term is often extended to suggest a mark, whether material or immaterial, that is evidence of something that has happened or has influenced a person or thing.
Vestige may be preferred to trace when the reference is to something that remains or still exists to give evidence of or testimony to the existence of something in the past; it often applies to remains (as a fragment, a remnant, or a relic) that constitutes a tangible or sensible reminder of what has gone before.
Track has come to be used more often than trace in the sense of a line of perceptible marks, especially in hunting, where it also may mean the scent followed by the hounds, and in geology, where it usually means a line of fossilized footprints.