Schiralli (2007) points out that our sense of the aesthetic is part of our ability to engage in formal scientific and mathematical endeavors:
"The identification of pattern is, therefore, a fundamentally
aesthetic apprehension that in systematic inquiry soon moves beyond
the immediately perceptable towards the more formal conceptual
connections with which scientific and mathematical theory is
ultimately concerned." (p. 106)
For a practical example, Frank Wiczek notes in the introduction to Weyl's (2009) Philosophy of Mathematics and Natural Science:
Time-reversal symmetry asserts the equivalence of past and future, in
the microscopic laws of physics. Of course, both the concrete history
of the universe and (at a mundane, but more specific and practical
level) the laws of thermodynamics distinguish past and future. Yet the
laws of Newtonian mechanics and Maxwellian electrodynamics--and
indeed, all the basic laws of the microcosmos known in Weyl's day--do
not. (p. xi)
Wiczek explains that symmetry has played a role in theory, and that violations of theoretical symmetry are important in that they have lead to questions that powered major advances in physics.
But maybe you wanted a more geometric example. Patterns in math are so often useful, and symmetry is no different. Lets say my pool is some strange shape. I want to calculate the volume of water in the pool so I know how much chlorine to use, but I don't have a formula for the shape in question. But then I notice that the pool is symmetrical in some way (choose whatever type of symmetry you prefer). The symmetry involves a shape I do have the formula for. So I calculate the volume of that shape, then iterate accordingly (or just multiply).
Perhaps you meant something different, but noticing symmetry in a system always gives you a way to reduce the complexity in how you represent the system. In the reverse, from a software or computer graphics standpoint, symmetry gives you a way to generate forms programatically by transformation of other forms.
Schiralli, M. (2007). The meaning of pattern. In N. Sinclair, D. Pimm, & W. Higginson (Eds.), Mathematics and the Aesthetic (pp. 105–125). Springer. Retrieved from http://link.springer.com/chapter/10.1007/978-0-387-38145-9_6
Weyl, H. (2009). Philosophy of Mathematics and Natural Science. Princeton University Press.