We humans memorize with our sight, just as we memorize with our hearing. Using one Mushaf helps us to get accustomed to the mental images of the pages, as well as the location of verses and their position on the pages of the noble Qur'aan. These images are printed on the mind as long as the person reads and looks at the same copy/version of the Qur'aan.
Version here means the type of print, and / or script of the Qur'aan. For instance, the Madeenah Mushaf is known to be exactly the same all over the world, but the Qatari print although very similar to it and having 15 lines per page also, is different from it in certain portions. Some prints of the noble Qur'aan make provision for the different types of recitation, others use different types of script (most commonly the Urdu script) and Allah knows best.
Every time we change the copy we are using, or even worse, change to a version/script of the noble Qur'aan that is alien to what we are used to, our memorization will be weakened. Recalling the verses from memory will be daunting and sometimes, can be impossible. This is because we will not have a fixed image (in our memory) of the page being recalled. Our memory of it will be scattered and haphazard, due to the change in mental impression. We get accustomed to whatever copy we use, the page order, the verse position, the hizb markings and so on. If we memorize from various versions of Qur'aan where the position of verses varies, we will find it difficult to memorize.
Scholars advise their students to have their own copy of the noble Qur'aan, and that they should not change it so as not to disturb their minds due to the differences in the page order and so on.
May Allah make this easy for us to stick to.
The source and main material used in preparing the content of this blog is the book, Ways of teaching the noble Qur'aan and memorizing it by Abu Abdrahman, Jamal ibn Ibrahim AlQirsh (طرائق تدريس القرآن الكريم وحفظه).