Concede and admit can overlap, but they are not exact twins.
Concede usually means accepting a point after resistance, debate, or pressure. It often suggests that someone gives ground reluctantly. You might concede an argument, a weakness, or that another person is partly right.
Example: She conceded that the deadline was unrealistic.
Admit means openly acknowledging that something is true. It is often more direct and commonly used for facts, mistakes, guilt, or personal feelings.
Example: He admitted that he had made a mistake.
A useful contrast is this: if the idea is I finally gave in and accepted your point, use concede. If the idea is I openly said this is true, use admit.
- Concede: The mayor conceded that the critics had a valid concern.
- Admit: The mayor admitted that the report contained errors.
In short, concede highlights yielding, while admit highlights acknowledgment.