I saw such usages many times(including one in this blog post I saw just now) especially when we are writing on the blackboard, but I don't know why between is simplified as b/w. It seems very strange to a non-native speaker when it meets the eye. Should logic be shelved in such casual cases? Any similar simplifications?
As with any other abbreviated expression, it depends entirely on whether your reader knows what you mean. When I see something new such "b/w", I have to evaluate it in the context of similar shorthand expressions. The slash commonly indicates two separate words, such as:
n/a = "not applicable"
i/o = "input (and) output"
y/o = "year(s) old"
and various others. My first guess therefore is that it is a variation on b&w, meaning "black and white". It may be that those who coined "b/w" to mean "between" confused it with "w/o" (meaning "without") and assumed any word can be shortened with a slash. However, "without" at least suggests two separate words ("with out") while "between" does not.
The point is that "b/w" does not fit standard shorthand conventions, so there is no reason why "between" is shortened that way. As others have mentioned, a simple dash, arrow, or slash would be sufficient to convey this meaning:
Visualize the relationship age - wage
Visualize the relationship age/wage
Still, once a certain shorthand becomes commonly used, it needs no reason for its existence.
It would probably be misleading to abbreviate "between" as "btw" since this shorthand already exists to mean "by the way".