Exodus exported private key
You are probably deriving a legacy P2PKH address (addresses that start with 1) versus the bech32 P2WPKH address (addresses that start with bc1) that Exodus uses.
A single WIF private key can generate multiple address types, other wallets and software are likely defaulting to a different type which is probably why you are having trouble. This is the same for seed phrases as well.
Also just FYI Bitcore-lib is an old outdated library and is not compatible with modern address types. The code example you provide would create a legacy P2PKH address (start with 1).