Well, looking back on my explanation, it's overly simplified. I'll elaborate:
Verkligen emphasizes the reality of what's stated. "Filmen är verkligen bra" means that "The movie was really good" in the sense that the reality of the quality of the movie is what's being emphasised.
Egentligen points out that something isn't what one might expect. "Jag är egentligen rörmokare" means that "I am actually a plumber" in the sense that one might not have though that.
Does that perhaps help a little bit?
Wow this comment thread is old so I doubt anyone is still paying attention.
I'm a nonnative Swedish speaker with about 15+ years of fluency with the language. I translated "Han är egentligen inte min bror" as "He is actually not my brother". A natural context for "Han är egentligen inte min bror" is one where someone thinks he is my brother (expects him to be), but I need to communicate to that person that he isn't. The Duolingo "best" translation to English seems strange to me. In English (my native language), if someone were to say "He isn't really my brother", the intended meaning would have to fit a context where he is pretending or claiming to be my brother, and I have to communicate to others that what he is claiming to be isn't the case.
"Verkligen" seems to me to involve an element of stress or emphasis on what is in fact the case.
I would translate "He is not really my brother" as "Han är faktiskt inte min bror", which carries the sense of asserting what is factually the case. This is how I differentiate "egentligen" from "faktiskt". I agree with Zmrzlina that "egentligen" is primarily used for the purpose of contraindicating an expectation.