The above quote by Albertine Gaur is the first line in his book A History of Writing. His premise is that writing approximates the original, human information storage - memory. I would like to expand that premise a bit to and point out that it also approximates speech and is therefore a communicative art as well.
Memory + Speech
Memory is only accessible to the owner, to the person who experienced it initially and remembers it. To share that with somebody who was not present at the impetus experience, a person must communicate. Communication through speech only gives a rough overview of the memory. How many times have friends told you a funny story and it wasn't funny? Even the reason for remembering is lost in the translation in that case. But even more basically, passing a memory to another means passing a memory to someone who doesn't have that memory, or the mindset, outlook, experiences, or intellect of the original. Speech loses a lot of the power of that memory.
Memory + Writing
Memory can be stored outside the body through writing. In writing it down though, the author is attempting to communicate to the page that memory through words. The author runs into the same problems that the speaker does above. The fullness, the context, the detail is lost and what gets communicated constitutes a new memory because it is so different that the original. However, through writing it down, the author gives up ownership of the memory to the device used to store the writing. Thereafter a reader may engage that written memory at the reader's own pace.
So in my mind, writing engages the same process speech does, but does it twice. Writing is not merely an approximation of memory, but also of communication - of making that memory useful. Where the speaker fills in details with inflection, hand gestures, and timing, the reader can travel at their own pace, can pause and do research or think about it, can take the time to form an imaginary mental picture. Writing and speech are both between memories, but writing itself constitutes a doubling which makes it its own memory, allows it to communicate years after the original experience had person have passed.