Same token, different question. The decoder shows you the raw JSON, which is what you want when copying a payload. The parser answers "what does this token actually say?" — every claim labelled with its registered meaning, and every timestamp converted from a Unix number into a date plus a relative age. Both share one decoding engine.
Which claims are standard?
RFC 7519 registers seven: iss (issuer), sub (subject), aud (audience), exp (expiry), nbf (not before), iat (issued at) and jti (token ID). Everything else is application-specific. The parser labels the registered ones and shows the rest verbatim.
My token says it is expired but still works. Why?
Expiry is only enforced if the verifier checks it. Some libraries require you to opt in to exp validation, and many allow a clock-skew tolerance of a minute or two. If an expired token is being accepted in production, that is a bug worth fixing in the verifier.