OpenSSL does not support any "BEGIN EC PUBLIC KEY" format and (AFAIK) no such format exists. There is a "traditional" public key format for RSA ("BEGIN RSA PUBLIC KEY"), which is essentially a PEM encoded PKCS#1 structure. Later standards defined how to encode public keys in an algorithm agnostic way (SubjectPublicKeyInfo) and so there isn't always a "traditional" format version.
You can check the known OpenSSL PEM types by looking in the openssl pem header file (include/openssl/pem.h). You will notice that there is no "EC PUBLIC KEY" entry. There is an "RSA PUBLIC KEY", "DSA PUBLIC KEY" and "ECDSA PUBLIC KEY", but the latter two are not used anywhere in the OpenSSL codebase (i.e. they only exist in the header file).
There are "traditional" formats for private keys. To convert a PKCS8 private key to traditional form use:
openssl ec -in p8file.pem -out tradfile.pem
The ec command also recognises the "-inform DER" and "-outform DER" options. So, for example, to convert a DER PKCS8 private key file into PEM traditional you could use:
openssl ec -inform DER -in p8file.der -out tradfile.pem
Or to convert a PEM PKCS8 private key file into DER traditional use:
openssl ec -in p8file.pem -outform DER -out tradfile.der
You can also go the other way, i.e. traditional form private key to PKCS8:
openssl pkcs8 -topk8 -nocrypt -in tradfile.pem -out p8file.pem
The pkcs8 command above also recognises the -inform and -outform options.