The two terms “obituary” and “death notice” are similar, but by definition, they refer to different things:
A death notice is a fairly short publication that offers the details of a death and provides information about funeral services and any other memorial activities. The family of the deceased and/or funeral home typically pays to have it published in the newspaper.
An obituary is a piece of writing, published after a death, that both provides details of the death and tells the story of the person’s life. It is typically written by a journalist and published in the newspaper alongside other articles written by journalists.
These two terms are often used interchangeably, and sometimes their usage is swapped. You might see the section of a newspaper with short, paid announcements of death referred to as the obituary section. Sometimes this is a casual reference by the newspaper’s readers, but sometimes even the newspaper itself calls this section the obituary section.
Similarly, you might see a short article that lists just a few basic facts referred to as a death notice, even if it was written by a journalist and wasn’t a paid announcement by the family of the deceased.
It’s worth knowing the definitions of these terms. But even if you don’t strictly adhere to those definitions, you’ll usually be able to get your point across, whichever you use.