Using SHA1 and RSA with java.security.Signature vs. MessageDigest and Cipher

OK, I've worked out what's going on. Leonidas is right, it's not just the hash that gets encrypted (in the case of the Cipher class method), it's the ID of the hash algorithm concatenated with the digest:

  DigestInfo ::= SEQUENCE {
      digestAlgorithm AlgorithmIdentifier,
      digest OCTET STRING
  }

Which is why the encryption by the Cipher and Signature are different.

To produce the same results:

MessageDigest sha1 = MessageDigest.getInstance("SHA1", BOUNCY_CASTLE_PROVIDER);
byte[] digest = sha1.digest(content);
DERObjectIdentifier sha1oid_ = new DERObjectIdentifier("1.3.14.3.2.26");

AlgorithmIdentifier sha1aid_ = new AlgorithmIdentifier(sha1oid_, null);
DigestInfo di = new DigestInfo(sha1aid_, digest);

byte[] plainSig = di.getDEREncoded();
Cipher cipher = Cipher.getInstance("RSA/ECB/PKCS1Padding", BOUNCY_CASTLE_PROVIDER);
cipher.init(Cipher.ENCRYPT_MODE, privateKey);
byte[] signature = cipher.doFinal(plainSig);