Certify, attest, witness, vouch are comparable when they mean to testify to the truth or genuineness of something.
Certify usually implies a statement in writing, especially one that carries one’s signature or seal or both or one that is legally executed; thus, a certified check carries the guarantee of a bank that the signature is genuine and that there are sufficient funds on deposit to meet it.
Attest (see also INDICATE) implies oral or written testimony from a person in a position to know the facts, usually but not invariably given under oath or on one’s word of honor; thus, when one says that something is well attested, he implies that there is sufficient documentary or oral testimony from competent persons to warrant its acceptance.
In technical legal use attest is used chiefly in reference to the official authentication of a document (as a will, a deed, or a record) or to the guaranteeing of the genuineness of a signature or a statement or an oath by a qualified public agent (as a notary public or a commissioner of deeds).
Witness implies attestation, not necessarily official or notarial, of a signature (as of a statement, a will, or a bond) by one who has seen that signature actually made and who subscribes his own name to the document as evidence of its genuineness.
Vouch (usually with for) rarely implies official or legal proof, which the other words in this group usually do imply, but it suggests that the one who testifies is a competent authority or a reliable person who will stand behind his affirmation and support it further if necessary.