The Seminole Tribe of Florida claims that they were kind of a breakaway faction that established a separate tribe.
Yeah, like I said, there is both kinship and animosity among the two tribes.
For example, if you ask Seminoles what "Miccosukee" means in their language, they will tell you "pig people." That's because the Spanish brought pigs to Florida with them and the feral pigs were caught and raised by Miccosukees -- and still today are a big component of their diet.
If you ask a Miccosukee what "Seminole" means in their language, they will say "the ones who ran away," meaning those who surrendered at the end of the Second Seminole War.
The truth is, both were part of the Creek Nation but have always spoken different languages and lived in different parts of the SE US.
The Lower Creeks spoke Mikasuky (a derivation of the extinct Hitchiti language) and lived mostly in GA and NE FL, so they had the closest contact with white settlers. In fact, they are responsible for the name "Creek," which is an English word, not any Indian language. The first contact, and longstanding friendly trade relationship whites moving into GA had was with the Indians who lived along a creek in NE GA -- so they were called the "creek Indians."
The ancestors of the Seminoles spoke Muskogee and lived further west and NW, mostly in more mountainous areas of NW GA and Tennessee. So, English settlers called the Indians they traded with "Lower Creeks," and those who lived in the mountains and wouldn't have anything to do with whites "Upper Creeks."
The languages of both groups come from the same roots. The languages share some words, but the words mean different things, and the languages are not mutually understandable. A person who speaks Mikasuky will not understand much of what a Muskogee-speaker says, so they communicate now in English.
The Miccosukees were forced to move south into Florida first as English settlement of Georgia progressed in the 1700's. The "Upper Creeks" didn't move to Florida in any numbers until the aftermath of the Creek War (1813-1814) that made Andrew Jackson famous. Many Upper Creeks were captured and sent to Oklahoma; the rest fled to FL, AL, MS, etc -- mostly FL because there were other tribes in the central Gulf states.
Both fought together tenaciously against the US in the Seminole Wars, and they are still close, but competitive. But they are actually two different groups. They are related, but separated by ancestry, language, and history.