Who knows the difference between 忘れてた and 忘れちゃった? In what kind of situations you use each of those?
In this case, it's the 6th definition ㋑ that ちゃう represents, "Implying that something unexpected happened".
So if you say 忘れてた it means, "I forgot". But わすれちゃった means "I forgot having unexpected consequences".
In my experience, using ～てしまう or ～ちゃう is more of a "whoops!" sort of comment, whereas 忘れてた is more of a statement of fact. "Do you remember coming here as a child?" would be more of the latter, while waiting in line for a baseball game and realizing you forgot the tickets would be more of the former 「チケットを忘れちゃった！」
In general, the use of either ～ちゃう or ～てしまう when referring to a personal screw-up is incredibly casual and not appropriate for work settings (you would generally apologize first, as it comes off a bit too light-hearted and as if you don't take the mistake seriously.
They're mostly the same, but 忘れちゃった has more of a sense of 'whoops' than just plain 忘れてた. I'd probably be more likely to use 忘れちゃった in a situation where the forgetting something had some kind of negative consequence (though probably a small one, as it's not a particularly formal form and it tends to have a sense of downplaying the consequences - if you really screw something up by forgetting something, you'd want to go the distance and say 忘れてしまいました.) It sounds kind of weird to use 忘れちゃった to mean something was forgotten but nothing bad came of it.
I think the important thing here is to understand the word しまう,and then to understand that ちゃう is the colloquial contraction of shimau. For simplicity's sake, I'll just write everything in romaji.
Shimau: Means to "completely" do something. Yatteshimau zo! means something like "I'm going for it!" It can have some sense of daring or regret associated with it. Little kids say "chau" all the time ("nichau yo", "I'll cry!), and parents use it to scold them: "Naicha dame da yo!" ("Don't you cry!"). In any case, shimau is used frequently (in both formal and informal speech and writing) and chau is also used frequently, albeit in speech. It is interesting to note that if you add the conditional "-ba" to the end of "-shimau" ("shimaeba") then it becomes more of an challenge or proposition of difficulty ("wasurete shimaeba" = "if you can manage to forget").
So, to answer your question, when used with wasureru, the only real way to interpret wasurete shimau is to see it as a statement of regret at having forgotten or unintentional forgetting, and "wasurechatta" means something similar although would not be said in quite the same circumstances.
This is in contrast to "wasureteta" which is a contraction of "wasurete ita" (sic "I had forgotten") which implies that you had forgotten but that someone or something just reminded you of that. The difference between "wasureta" and "wasureteta" is like the difference between "I had forgotten" (which implies that you now remember) and "I forgot."
忘れてた means 'I forgot', but it carries a sort of 'continuous' meaning, i.e. somebody has forgotten something for a period of time. At least that's how I've come to understand it.
On the other hand, 忘れちゃった is a colloquial, shortened version of 忘れてしまった.
～てしまう is a verb-only conjugation and has three basic meanings, which are:
- The action expressed by the verb was done unintentionally, by accident.
- The action was done completely, finalized.
In this case, 忘れちゃった may mean both, as in:
Whoops, I totally forgot.