Is the difference purely legal or is there a technical difference between a notary notarizing the document signer's identity and a certification authority certifying a signer's certificate?
A certificate authority will generally only sign certificates of identity containing public keys in a format intended to be interpreted primarily by machines. A digital notary or eNotary uses digital signature technology to notarize documents where the contents are intended to be interpreted by humans. That may include notarizing of digital identity documents eg a PDF image of your passport. While A CA might issue PDF-signing certificates they won't sign PDFs themselves in their capacity as CAs only identity documents for those who do sign PDFs.
I am not an a lawyer, but I imagine you're dealing with two different systems; the internet (policies set by tech companies), and the law (policies set by governments and courts):
certify the identity of a document signer. By following polices set by the CA/Browser Forum (a consortium of tech companies) and being included in the trust stores of Chrome, Firefox, Windows, etc, CAs have the power to declare that a certain public key belongs to a certain entity (person or machine) within the context of the internet.
The objective is for the signature to be verifiable by software -- depending on the type of document, that software may be a web browser, a PDF viewer, or the OS when you are installing software.
A notary is a person licensed by the government to perform acts in legal affairs, in particular witnessing signatures on documents.
So they also certify the identity of a document signer. By following local laws and obtaining a notary licence from their government, notaries have the power to declare something as being a legal document within the context of the country's laws.
The objective is to certify that a given document was drafted and signed in accordance with the country's laws, and that the signers are who they say they are such that it can be admissible in court.
Notaries may employ digital signature technologies, from wikipedia
An eNotary is a Notary Public who notarizes documents electronically. ... Electronic notarization is a process whereby a notary affixes an electronic signature and notary seal using a secure Public key to an electronic document (such as a PDF or Word document).
The details around eNotaries will vary from country to country -- that page contains a list of countries with laws around eNotaries. More generally, see the list of countries with laws around digital signatures.
I agree with you that core to both CAs and notaries is the issue of verifying the identity of the signer, and both may involve cryptographic public keys. In general though, the policies governing CAs (CAB Forum) are completely separate from the policies and laws governing notaries (country-specific) so being one does not imply that you are the other.
I am a Colorado notary, electronic notary (e-notary), and a certified notary training instructor. I also use PGP keys for encrypting and signing email.
The primary purpose of a notary public is to prevent fraud by verifying the identity of a legal document signer. We also observe for document understanding, mental incapacity, and coercion. For some documents, we place the signer under oath to tell the truth, under criminal penalty of perjury.
Notary law defines which forms of identification are acceptable. If all conditions are met, we attach a completed notarial certificate including the location (state & county), date (not time), verified name of signer, type of notarial act, notary name and notary stamp. A notary is a commissioned public official, so it is an official government stamp, recognized by the law as true and correct on its face, prima facie evidence.
The document includes the handwritten signature of the customer and the notary.
As an electronic notary, I can sign an electronic document with any form of electronic signature allowed by state law. Note that an electronic signature is applied by electronic means, but is not limited to a digital signature using cryptography. An image file of a hand-written signature, attached by the principal, is also a type of electronic signature.
State laws vary on the method used by an e-notary to attach an official notary stamp electronically. In Colorado, we can use a randomized verifiable Document Authentication Number (DAN), issued by the Secretary of State to a specific notary. No specialized software, hardware, or public key is needed. The customer usually uses Adobe software that accepts electronic signatures. The document is "sealed" to prevent tampering once the notary applies the e-signature and e-stamp.
A Certificate Authority (CA) may or may not verify the identification of the human certificate owner, depending on the type of digital certificate issued. Since a CA is a business, not a government agent, it cannot attach a government notary stamp.
A notary public does not issue digital certificates used for cryptography. A notary public is a public official and must be a human that obeys notary laws.
A Certificate Authority does not notarize documents as a government agent. A Certificate Authority is not authorized to place a human under oath to tell the truth, under penalty of perjury. A Certificate Authority can use a human operator or software to issue digital certificates.