I heard from another user that "ada" and "berada" mean the exact same thing ("to be, to exist"). I think he said that "berada" is a tad more formal, though.
It is correct, and more commonly used in daily conversations. "ber-" is usually left out in daily conversations. But if you're writing a formal essay, using "ber-" is better.
in addition I believe that ber- is used to transform an noun into a verb, like with usia for example: Usia saya dua puluh tahun. - - > Saya berusia dua puluh tahun. :)