See https://polarssl.org/kb/cryptography/asn1-key-structures-in-der-and-pem (search the page for “BEGIN RSA PRIVATE KEY”) (archive link for posterity, just in case).
BEGIN RSA PRIVATE KEY is PKCS#1 and is just an RSA key. It is essentially just the key object from PKCS#8, but without the version or algorithm identifier in front. BEGIN PRIVATE KEY is PKCS#8 and indicates that the key type is included in the key data itself. From the link: