These are two of Aristotle's three modes of persuasion. Ethos persuades through the character and credibility of the speaker — why you should trust them. Logos persuades through reasoning and evidence — why the argument holds. The third is pathos, which persuades through emotion. Most effective arguments use all three.
The core difference
- ethos — credibility. "As a surgeon of thirty years…" The audience is being given a reason to trust the speaker.
- logos — reasoning. Evidence, data, logical structure. The argument stands on its own merits.
- pathos — emotion, included here because the pair only makes sense as part of the trio.
The everyday meaning of ethos
Outside rhetoric, ethos has taken on a second life meaning the characteristic spirit of a group — "the ethos of the school". That sense is entirely standard and is far more common in ordinary use than the rhetorical one, which is worth knowing before assuming a writer means Aristotle.
Logos elsewhere
Logos is Greek for word, reason or account, and carries theological weight in Christian usage. It is also the root of logic and of the -logy ending in every field name from biology to etymology. It is not the plural of logo, which is logos by coincidence of spelling only.