All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The word As-Samaa’ (heaven) is the singular of As-Samaawaat. As-Samaa’ may refer to the genre (i.e. it is a generic noun in the sense that it refers to something in general), so its meaning would be plural (heavens), like the saying of Allaah (which means): {Then He directed Himself to the heaven, [His being above all creation], and made them seven heavens.} [Quran 2:29]
As-Samaa’ Ad-Dunya, is the nearest heaven to us, in comparison to what is above it. In the Quran, it is mentioned three times in the context of it being adorned with stars and planets, and all of these are only in the nearest heaven to the earth, as Ash-Sha’raawi said in his Tafseer (interpretation of the Quran).
On the other hand, there is no difference between the verse in Chapter Aal-Imraan [Quran 3] and the verse in Chapter Al-Hadeed [Quran 57]; rather, they have the same meaning. Shaykh Muhammad Ameen Ash-Shinqeeti said in his Tafseer entitled Adhwaa’ Al-Bayaan: “The verse in Aal-Imraan explains that what is meant by heaven (As-Samaa’) in the verse of Chapter Al-Hadeed is its genre which means all the heavens, as it is obvious, and Allaah knows best.”
As regards what is meant by what is between the heavens and the earth, it is this vast universe which is between both of them. And what is between them are the different kinds of creatures such as the jinn, humans, birds and other creatures.
However, we have not come across any special statement of the scholars about the relationship between these terms and what you called our present scientific understanding of the universe.
Finally, it should be noted that the Quran is a book of guidance which leads the people to the worship of Allaah, and it is not a book of scientific theories, as the Quran was not revealed for this purpose. Rather, it is a methodology to adjust life and society according to the relationship between the people and their Lord The Almighty, and according to the conduct that their Lord is pleased with.
Nevertheless, the Quran refers sometimes to some universal facts to prove the existence of the One true God, Who is able to do everything.
Allaah Knows best.