The 30 second answer
Basic sends an encoded username and password, while Bearer presents a token whose possession grants the represented access. Neither encoding nor the word token makes transport safe, so both need TLS. I prefer short-lived, appropriately scoped tokens where the architecture supports them, and I avoid logging credentials. A bearer token also needs validation and authorization checks, not just extraction from a header.
HTTP Basic constructs credentials from a user identifier and password and Base64-encodes them. Base64 is reversible encoding, not encryption. Basic does not itself provide confidentiality, and repeated use exposes the underlying credential to each accepting endpoint unless protected by the surrounding system.
The Bearer scheme carries an access token, typically in the Authorization header. A bearer token can be opaque or structured; JWT is not required. Whoever possesses it can use it within its accepted scope and lifetime, so theft is a serious risk even when the user's password is never sent to the resource server.
I would require TLS, redact authorization headers, and define token expiry, audience and revocation or replacement policies. Sending tokens in URLs can leak them through logs or other URL exposure and should be avoided. The decision depends on client type and the authorization system, not on a claim that Bearer inherently prevents replay. An expired or wrong-audience token must be rejected even if its structure appears valid.