The outer signature protects the ciphertext against change. If the signature is verified before decryption then the signature can for instance protect the ciphertext against padding oracle attacks, both on the asymmetric encryption as well as the symmetric encryption that is commonly within a hybrid cryptosystem.
The encryption of the plaintext and inner signature of course offers confidentiality of the message (see the last section why the signature itself also needs to be encrypted).
The inner signature protects the integrity of the message after decryption. Obviously the outer signature (over the ciphertext) cannot be used on the plaintext message.
So you would either have to store the ciphertext (which means you'd have to decrypt the message with your private key each time to view the verified contents) or you would loose the ability to verify the plaintext message.
However, if you sign the plaintext message as well then you can still verify the message using the sender's public key even after decryption.
Note that encrypting the inner signature prevents an attacker of retrieving information about the plaintext. If it was directly available then the attacker could verify hashes over possible plaintext using the public key of the sender.
For instance, an attacker could verify that the message is "Yes" or "No" by guessing either message and then verifying the signature.