I've read a number of questions on here regarding のだ, but what I'm looking for in this question is the nuance のだ imparts to a sentence as compared to when a similar construction is used. There are two usages of it by which I am particularly befuddled.
When のだ is used to provide an explanation, how does it differ from (だ)から? For example, how do the following two sentences differ? (And no, I'm not asking about the difference between のだから and だから; I've found satisfactory answers on that topic.)
今は行かなければいけない。母からの電話がきたから。
今は行かなければいけない。母からの電話がきたんだ。
When occurring as のだ{が/けど}, how does the sentence differ from when the の is dropped? The answer given to this question describes this usage as providing background, but can background not be provided without the の? And again, how do the following two sentences differ?
宿題をしたんですが、学校に持ってくるのを忘れました。(taken from here)
宿題をしましたが、学校に持ってくるのを忘れました。