That is a really good question, and the answer is that nobody really knows for certain.
The whole concept of duality was deeply ingrained into ancient Egyptian culture. The "Black Land", kmt or "Khemet", (a reference to the fertile strip along the Nile), and the "Red Land", dšrt or "deshret", (referring to the deserts beyond that fertile strip) is one example. The two kingdoms, Upper Egypt and Lower Egypt, united under a single pharaoh is another.
As you say, the two crowns were associated with the two kingdoms. The Red, "Deshret", crown was associated with Lower Egypt, and the White, "Hedjet", crown with Upper Egypt.
There was also the "Double Crown", or "Sekhemti", which represented the unified kingdom, and incorporated both the Red Crown of Lower Egypt and the White crown of Upper Egypt.
(There were also other crowns with specific associations and meanings)
Pharaohs are portrayed wearing all three of these crowns right up to the Graeco-Roman period in Egypt, although as far as we know, there is generally no obvious reason to show why any Pharaoh was portrayed wearing a particular crown in an inscription. We were taught that the crowns continued in use in royal portraiture to emphasise the dual nature of the two lands.
(There are exceptions to this, however, as in the case of the Pharaoh Ahmose, who expelled the Hyksos from Lower Egypt, portrayed in the Hedjet Crown of Upper Egypt)
It is by no means certain that any of the crowns were actually worn by any pharaoh of the later periods. None of them survive (even in the (relatively) untouched burial of the young Tutankhamen). It may be they were traditional symbols, preserved in the iconography in inscriptions at temples, but never actually worn.
Sources:
- Allen, James P: Middle Egyptian, Cambridge University Press,
2000
- Bard, Kathryn: An Introduction to the Archaeology of Ancient
Egypt, Wiley, 2008
- Kemp, Barry J: Ancient Egypt: Anatomy of a Civilisation,
Routledge, 1991
- Wilkinson, Toby A H: Early Dynastic Egypt, Routledge, 2002
