のだ vs. similar constructions


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.

  1. 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** 。

  2. 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](https://japanese.stackexchange.com/questions/3349/how-is-the-%E3%81%AE%E3%81%A7%E3%81%99-working-here) 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](https://j-nihongo.com/ndesuga/))
>
> 宿題をしましたが、学校に持ってくるのを忘れました。




> When のだ is used to provide an explanation, how does it differ from (だ)から?

Giving a reason is only one of the ways to explain something. Explanatory-のだ
is used in [broader situations](https://www.wasabi-jpn.com/japanese-
grammar/explanatory-noda/) to mark an important part of your conversation.
When it does provide a reason for the previous sentence, it's semantically
interchangeable with から. The difference is から is more explicit, unambiguous
and matter-of-factly. から is preferred in logical sentences such as math
proofs. In casual and/or lively conversations, から is less common. You can even
combine both から and ん and say 電話が来たからなんです. This is more emphatic and vivid
than plain 電話が来たからです.

> 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。  
>  宿題をしましたが、学校に持ってくるのを忘れました。

This is a typical explanatory-の which provides a clarification of the
_previous_ context. A sentence like this is usually used as an excuse for a
question like "Why did you forget your homework?". In such cases, the first
sentence should be used. Without の/ん, it doesn't sound like an excuse at all.
If this was an answer to a question like "How was your last vacation?", the
neutral second sentence could be fine.

