＊＊＊ 忘れて（い）た： I had been forgetting (but now I remember). OK: ヤバイ、今日は結婚記念日だ。すっかり忘れてた。 NG: ヤバイ、今日は結婚記念日だ。すっかり忘れちゃった。 ＊＊＊忘れちゃった　＝　忘れてしまった。 ***1: I have forgetten/ (have happened to forget), (and now I don't remember). OK:　あれ、いつだったっけ？ 忘れちゃった。思い出せないや。 NG:　あれ、いつだったっけ？ 忘れてた。思い出せないや。

They can covey same meaning, and both say "I forgot (something)" basically. However, 忘れちゃった is a casual expression. Children often say to their mothers "……するのを忘れちゃった" as an excuse. If I say "忘れちゃった" to my boss, he is going to be angry at me, because it is a casual and cute expression, which most people mainly use to their relatives and their close friends.

忘れた and 忘れちゃった basically means the same, but ちゃった implies the idea of "I didn't mean/want to, but it happened". 忘れてた means that you forgot something and it has been in that state for a while. If you use ちゃった for this, then you can say 忘れちゃってた.