jwt.io, run by Auth0/Okta, is the best-known JWT debugger and can verify a signature against a pasted key. The LocalFirstTools JWT Decoder only decodes the header and payload, never verifies, and has no field for key material. Use jwt.io when you need signature checks. Use the local decoder for quick claim checks.
The short verdict: jwt.io (run by Auth0/Okta) is the canonical JWT debugger and an excellent tool; if you need signature verification with a pasted key or its library directory, use it. The LocalFirstTools JWT Decoder is the simpler choice when you want fast decoding inside a broader local-only toolkit, with no vendor relationship and a hard rule that key material never belongs in the page.