Choson (or Joseon in more modern spelling) was a historical name for korean kingdoms. The first was Go-Joseon, "Go" meaning "ancient". The latest was Joseon, sometimes called "Yi Joseon" (Yi being the dynasty of kings who ruled it) to distinguish it from the ancient one.

Colloquially it's still used for "korea" in the south for a few rare things., quick example that comes to mind is "Joseon O-ee", literally "Joseon Cucumber", to refer to korean cucumbers. Joseon-jok, referring to the ethinically korean people in southern manchuria, is another term where Joseon is used because it's a neutral term for "korean" that has no political affiliation.

"Hanguk" is a shortening of the formal name of the Republic of Korea, "Dae Han Min Guk" literally "Greater Han People's Nation". The "Han" is different from the chinese han btw. It's a name that was coined by the provisional government in Shanghai that opposed the japanese occupation. Same with the flag, slightly modified, that the ROK uses.

Nowadays nobody uses "joseon" for korea as a country in the south, you're typically looking at very old people, "hanguk" is just more familiar with younger generations, and there's negative connotations from using the same terminology as the north.

Both still claim to be the legitimate government of the entire country. At no point was the division ever accepted by either side. For instance, both have provincial government structures for the parts of the country outside their control.