These words are related, but their meanings are quite distinct.
Interment means “burial”—the event of placing a dead person in a tomb or grave; it is a more formal word than burial. Here's interment in a sentence:
The interment of the politician’s body was attended by thousands of people.
Internment means the condition of being a prisoner in an internment camp—a place for holding prisoners of war and other people considered dangerous to a nation, such as the internment camps for Japanese people in America during WWII.
Internment camps are often compared to concentration camps—the places where the Germans imprisoned Jews and gypsies during WWII. However, this is incorrect. You can only say concentration camp, if the purpose of the place is genocide (murdering a racial or ethnic group). Internment camps are only for holding people, though conditions can still be brutal and inhumane.