Swear
To make a solemn promise; to vow, usually before God. Jurat wording includes the phrase, "Subscribed and sworn to before me ..." Notaries routinely have signers and witnesses swear under oath. Credible witnesses swear to the identity of a signer. Signers often have to swear to the truthfulness of a document.
Sworn Statement
A sworn statement requires the affiant to raise their right hand and swear under oath or affirmation that the statement they made is true. The affiant could also swear that the contents of a document are true and correct. The notary public might ask an official question such as, "Do you solemnly swear that the statements made in this document are true and correct?".