Difference between RSA key "BEGIN RSA PRIVATE KEY" and "BEGIN PRIVATE KEY"

BEGIN RSA PRIVATE KEYIt's PKCS#1, which is just an RSA key. It's essentially just a key object in PKCS#8, but without a version or algorithm identifier in front. BEGIN PRIVATE KEYis PKCS#8, indicating that the key type is contained in the key data itself. From the link:

Unencrypted PKCS#8 encoded data begins and ends with tags:

-----BEGIN PRIVATE KEY-----
BASE64 ENCODED DATA
-----END PRIVATE KEY-----

In the base64 encoded data, the following DER structure exists:

PrivateKeyInfo ::= SEQUENCE {
  version         Version,
  algorithm       AlgorithmIdentifier,
  PrivateKey      BIT STRING
}

AlgorithmIdentifier ::= SEQUENCE {
  algorithm       OBJECT IDENTIFIER,
  parameters      ANY DEFINED BY algorithm OPTIONAL
}

So for an RSA private key, the OID is 1.2.840.113549.1.1.1 and there is an RSAPrivateKey as the PrivateKey key data bitstring.

In contrast BEGIN RSA PRIVATE KEY, it always specifies an RSA key and therefore does not contain the key type OID. BEGIN RSA PRIVATE KEYis PKCS#1:

RSA private key file (PKCS#1)

RSA private key PEM files are specific to RSA keys.

It starts and ends with tags:

-----BEGIN RSA PRIVATE KEY-----
BASE64 ENCODED DATA
-----END RSA PRIVATE KEY-----

In the base64 encoded data, the following DER structure exists:

RSAPrivateKey ::= SEQUENCE {
  version           Version,
  modulus           INTEGER,  -- n
  publicExponent    INTEGER,  -- e
  privateExponent   INTEGER,  -- d
  prime1            INTEGER,  -- p
  prime2            INTEGER,  -- q
  exponent1         INTEGER,  -- d mod (p-1)
  exponent2         INTEGER,  -- d mod (q-1)
  coefficient       INTEGER,  -- (inverse of q) mod p
  otherPrimeInfos   OtherPrimeInfos OPTIONAL
}