Atlantis is a mythic civilisation lost underwater according to European mythology.
The Continent of Mu is a mythic civilisation lost underwater according to pre-columbian South American mythology.
The myths of Mu were researched and published by Churchward in the 1920s.
Are there any common details between the two lost civilisations?
The biggest one, to my mind, is that neither of them are really mythological, per se.
Atlantis was a fictional island made up by Plato. It serves as the fictional antagonist to Plato's ideal state in "Timaeus" and (especially) "Critias". They are written in the dialectic style, in which the character Critias narrates the story of Atlantis.
Scott Arthur Leonard provides a statement on it's classification in his article "A Closer Look: Is Plato’s Story of Atlantis a Myth?"
It is, perhaps, most accurate to conclude that Plato’s Atlantis story is not myth so much as it is mythic. That is, it imitates myth in most important respects. It projects metaphysical assumptions, it comments on cultural origins, and dramatizes a set of virtues, values, and beliefs just the way a better-developed, more culturally representative myth might do.
And a more thorough treatment of it's fictional nature can be found in Diskin Clay's "Plato's Atlantis: The Anatomy of a Fiction"
There remains debate on whether there was some other story (mythological or historical) that inspired Plato's island.
The continent of Mu sprang from a discredited translation of a Mayan Codex. From there, it was developed into a full-blown hoax by James Churchward.
The Codex in question was known as the Troano Codex, but has since been combined with the Cortesianus Codex, and is known (collectively) as the Madrid Codex. It's actual contents, having since been better translated, are summarized in "Research Methodologies and New Approaches to Interpreting the Madrid Codex" by Gabrielle Vail and Anthony Aveni:
...it is the longest of the surviving Maya manuscripts, containing approximately 250 almanacs concerned with a variety of topics, including rain ceremonies associated with the deity Chaak, agricultural activities, ceremonies to commemorate the end of one year and the start of the next, deer hunting and trapping, the sacrifice of captives and other events associated with the five nameless days (Wayeb’) at the end of the year, carving deity images, and beekeeping.
