The default format of keys was changed in OpenSSL 1.0. From OpenSSL 1.0 change log:
Make PKCS#8 the default write format for private keys, replacing the traditional format. This form is standardised, more secure and doesn't include an implicit MD5 dependency. [Steve Henson]
Good explanations of the difference between the two formats: https://tls.mbed.org/kb/cryptography/asn1-key-structures-in-der-and-pem
Converting RSA private key:
To convert from PKCS#8 to PKCS#1:
openssl pkcs8 -topk8 -inform pem -in file.key -outform pem -nocrypt -out file.pem
To convert from PKCS#8 to PKCS#1:
openssl rsa -in key1.pem -out key2.pem
Converting RSA public key:
To convert from PKCS#8 to PKCS#1:
openssl rsa -pubin -in <filename> -RSAPublicKey_out
To convert from PKCS#1 to PKCS#8:
openssl rsa -RSAPublicKey_in -in <filename> -pubout
Converting EC private key:
To convert from PKCS#8 to PKCS#1:
openssl ec -in ec2.pem -out ec3.pem