I think they have close meaning.
運良く is more objective than ありがたいことに.
運良く is just merely "by good luck". It is just luck. Or "by good chance." It is just by chance.
ありがたいことに is more subjective including the emotion of thanks.
Thanks to someone, or thanks to God.
運良く is a little more formal and preferred in a little more objective description (not completely objective, of course, talking about luck).
ありがたいことに is a little more colloquial and preferred in a little more subjective speech.