From Merriam-Webster:

Funeral Director

1: a person whose job is to arrange and manage funerals

Mortician

1: a person whose job is to prepare dead people to be buried and to arrange and manage funerals

Undertaker

1: one that undertakes: one that takes the risk and management of business; entrepreneur

2: one whose business is to prepare the dead for burial and to arrange and manage funerals

The titles of funeral director, undertaker and mortician all apply to someone who manages a funeral, however there are a few differences between the titles. A funeral director typically specializes in meeting with families to organize and make arrangements for the final placement and funeral service details. Someone who refers to themselves as an undertaker often specializes in preparing a body and performs the burial. The term undertaker started as a euphemism. The term mortician is generally used to refer to a person who both makes funeral arrangements and prepares the body for burial or interment.