What advantages does a confidential marriage have?

No, your decision to change or hypenate your name or keep your name is independent of whether your marriage is confidential or not.

You have to be living together as spouses at the time you apply for a confidential marriage license, and must sign an affidavit on the license attesting to it. You have to be married in LA County if LA was where the license is issued. No witnesses are required at the ceremony, and no witnesses sign the marriage license.
The confidential marriage license is a confidential record registered at the County Clerk’s Office. Only the couple can buy copies of the marriage license and have to show valid picture ID. Anyone else who wants a copy needs a court order.

Besides more privacy, one of the advantages of a confidential marriage license is that a specialized notary can provide an immediate signed and notarized duplicate of your marriage license that's valid proof of your marriage for all federal and state governmental purposes, such as immigration issues or property issues. If you marry with a marriage license issued by any other county besides LA, you have to wait up to 10 weeks or more to receive a certified copy of your marriage license and certificate.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.