One example, slightly outside of mathematics, is the Feynman diagram which represent the interaction between particles in a field over a period of time. They're slightly different from space-time diagrams, but make it easier to see the contributions of particles and anti-particles to an interaction.
Another example, in chemistry and mathematics, is that a series of chemical reactions can be described by a series of equations
- $A + B \to C + D$,
- $D + E \to F + G$,
- $G + H \to J + A$
which does not clearly describe the "ins and outs" of this chemical cycle as well as a graph (directed graph) diagram does:
: A : J-<-- / \ --<--B : \/ \/ : H-->/ \---> C : / \ : / \ : G-----------D : / \ : F <-- -<--E
which clearly illustrates that the cyclic nature of this series of actions or the catalytic nature of some of these moities.
Sometimes, rewriting the steps of a proof simplifies understanding of it at an earlier level of our education, while it seems wholly un-necessary at later parts of our education. For example, $(p+q)^3$ is easily expanded in our heads to 1, 3, 3, 1, and $p^3, p^2q, pq^2, q^3$ and recombined into $p^3+3p^2p+3pq^2+q^3$, even though the initial and final $1$ of the binomial coefficients are effectively silent. But for beginning algebra students in high-school, putting the ones back into the expansion makes it easier to comprehend.
In the general case about encountering a new notational technique the first time may simply be about how the mind deals with novelty. In the example of Willie Wong about the Grothendieck's school's vertical arrows, does it really bring in a different mode of thinking, or does it merely indicate that the vertical arrows are viewed as being different from the horizontal arrows, and that this difference is viewed as representing a different mode of action?
In other words, unfamiliar objects or symbols are viewed as being novel, exciting certain parts of the brain more than familiar objects would. Familiar objects are already "wired up", whereas novel objects are not pre-conditioned to elicit a particular point of view, thus allowing a different approach to be considered.