Trisavó vs. tataravó
Both trisavó and tataravó refer to distant female ancestors, specifically around the level of a great-great-grandmother. The primary difference lies in regional usage and technical accuracy: trisavó is the technically correct term used in genealogy and Portugal, while tataravó is typically widely used in Brazil for the same relative, even though it technically refers to one generation further back.
Trisavó
C1This is the morphologically accurate term for a great-great-grandmother (the mother of a great-grandmother). It uses the prefix tri- to indicate she is the third grandmother within the lineage. It is the standard term used in Portugal and in formal genealogical contexts in Brazil.
A minha trisavó nasceu na cidade do Porto em 1890.
(My great-great-grandmother was born in the city of Porto in 1890.)
Para completar a árvore genealógica, preciso do nome da sua trisavó.
(To complete the family tree, I need your great-great-grandmother's name.)
Em Portugal, as pessoas dizem trisavó logo depois de bisavó.
(In Portugal, people say great-great-grandmother right after great-grandmother.)
Tataravó
C1In Brazil, this word is commonly used in everyday speech to mean great-great-grandmother, acting as a direct synonym for trisavó. However, historically and technically, it is a variation of tetravó (great-great-great-grandmother). While the dictionary definition points to the fourth generation, most Brazilians use it for the generation immediately following the great-grandmother.
O meu filho conheceu a tataravó dele antes dela falecer.
(My son met his great-great-grandmother before she passed away.)
No Brasil é muito comum chamar a mãe da bisavó de tataravó.
(In Brazil, it is very common to call the great-grandmother's mother the great-great-grandmother.)
Nós encontramos uma carta antiga que pertencia à minha tataravó.
(We found an old letter that belonged to my great-great-grandmother.)
Summary
In short, if you want to be technically precise or are in Portugal, use trisavó for a great-great-grandmother. If you are in Brazil speaking informally, tataravó is the most frequently heard term for that same relative, even though it technically implies one generation further back.