Both sentences sound correct to me, but one is more grammatically correct. Which one and why?
The room felt cold because we didn't close the windows.
The room felt cold because we hadn't closed the windows.
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The difference between these two is actually that the "didn't" version is what is called the simple past and the "hadn't" version is called the pluperfect. In practice, particularly in informal English, there is very little difference between the two and many native speakers will use them interchangeably.
The difference between the two is that the pluperfect emphasizes that the action you're talking about (closing the window) happened before the other thing you're talking about in the past. It could be used to contrast that afterwards (when you noticed it was cold), you closed the window, whereas the simple past very slightly infers that the window was still open even after this.
"The living room was cold, because we hadn't closed the window; so I closed it and lit a fire in the fireplace."
"The living room got wet in that storm, because we didn't close the window."
Another possible distinction between them is that "we didn't close the windows" could be taken to mean a "habitual" action that was true of this past time (when you were cold): that is to say that at this time you are speaking about, you commonly left your windows open, and thus the room was cold. "Hadn't closed the windows" doesn't carry that same potential meaning.
An example of this usage:
"When we lived at that house on Smith Street, the living room was always freezing cold, because we didn't close the windows!"
Personally I would use "The room felt cold because we hadn't closed the windows."
I'm not 100% sure why this sounds better but I think it might be because it implies a step was missed (i.e. closing the window), and that was the cause of the coldness.
I think "The room felt cold because we didn't close the windows." implies a concious decision not to close the window.
Having said that I think both are perfectly correct grammatically and context would probably dicate which one was used.