Usually a shriek is smaller/quieter than a scream. A shriek is also high-pitched, while a scream may not be. People usually shriek when surprised (ex: something jumping out at you, jump scare in a movie, etc). A scream is expresses extreme emotion, usually fear. Screams are also usually longer, while shrieks are short. Examples: She shrieked when she saw the spider on the floor. She screamed in the haunted house because she is terrified of clowns.

In addition to the answer above, people can scream at each other when they are angry. Shrieks and usually one-off, without words. Screaming can include words, and also carries the meaning of “talking very loudly”, especially when angry.