The word berichten is only used in an official-ish context. Examples:
Berichten Sie mir bitte bis morgen, was Sie letzte Woche erreicht haben! (context: boss’ speech at work.)
Der Sachverständige berichtete … (translation: reported)
It always implies leaving out the unimportant part and saying nothing but the truth.
The word erzählen mainly just means telling something that will likely be a story at the end.
Erzähl mir von deinem Wochenende!
Erzählst du mir eine Gute-Nacht-Geschichte?
Whether the story is true depends on the context; telling about the week-end implies truth while the bedtime story doesn’t. But likely you wouldn’t leave out all too much, even if it’s unimportant.
