Essentially, yes. A funeral director is an undertaker. The terms are often used interchangeably but usually refer to the same thing.
The term undertaker was first used in mediaeval Britain when they were known as funeral undertakers. Woodworkers were the first funeral undertakers, as skilled carpenters they were able to make coffins. If they had transport, a horse and cart, they would also transport the body, from the deceased’s home, most people died at home then, and would take them to the local cemetery.
Over time these woodworkers took on more responsibility and would care for the body and organise the funeral service. It is not known exactly when the term Funeral Director was first used, but in 1905 the British Undertakers Association became the National Association of Funeral Directors (NAFD) and it is thought to have been around this time.
The change from the name undertaker to funeral director possibly came about as more responsibility was involved in directing every aspect of the more modern funeral.
Firstly by bringing the person who died into their care and looking after them until the day of the funeral.
Meeting with the family or friends of the person who has died to discuss your plans and ideas for the funeral or memorial service or ceremony.
Support in the completion of paperwork and documentation.
Arranging the venue for the funeral and helping to find the right person to lead the service or ceremony on the day of the funeral.