I want to know what the difference is between ils sont and ils s'appellent. One is the part of être and other is s'appeler. Both expressions mean more or less “they are”. In what way are they different?
Actually, translating the same way in English is almost all these two expressions have in common. The verbs you rightly identified bear different meanings (as Stéphane mentions, to be and to call oneself), both incidentaly covered by are.
Ils sont means “they are.” Ils s'appellent means “they are called.” It is the “called” part that makes the second a bit more restrictive, and separates one from the other.
They are the same in some aspects but there are some differences,
Nous avons deux verbes différents ici, être et s'appeller.
Here is an example:
Ils sont Paul. -- They are Paul. Ils s'appellent Paul. -- They are called Paul.
There is a group of people who are male and named Paul.
Both of them are same but with a different syntax.