見える：There is no obstacle, such as a wall, cloth, distance, fog etc., between the viewer and the thing. It doesn't matter whether the viewer wants to see it or not. 見られる：The viewer has a chance to see. A: きのうは月食（lunar eclipse）でしたね。 B:そうだったんですか。知りませんでした。見に行きましたか？ A:見に行ったんですが、空がくもっていて、よく見えなかったんです。 (I went to see the moon, but I couldn't see it clearly because the sky was cloudy.) B：そうですか、残念でしたね。 A：ええ。でも、月食は来年また見られますからね。 (We will have another chance to see the lunar eclipse next year.) The same explanation is applied to 聞こえるand 聞ける. 聞こえる: There is no obstacle, such as noise, wall, distance etc., between the listener and the sound. IT doesn't matter whether he/she wants to hear it or not. 聞ける: The listener has a chance to hear.

"Mieru" means that something is visible, whereas "mirareru" is used for things that take an act on your part to see, or actively watching something. Here's a better explanation. Look for the heading that says "Are 「見える」 and 「聞こえる」 exceptions