First for it's importance in the field of abstract algebra:
This function returns the cardinality (order) of the group $U(n)$ closed under modular multiplication.
This function also returns the upper bound for the order of an arbitrary element in the $U(n)$.
And after that it's computational importance in modular arithmetic:
Since residue exponention is not well-defined, reducing the exponent modulo $\varphi(m)$ is our only way out for simplyfing modular exponents due to the Euler-Totient theorem.
With the same way above, if exponent of the element is relatively prime with the $\varphi(m)$, we can compute modular inverse of the exponent and with the help of it we can calculate modular roots. It is essential to RSA.
Are my points true? What i can add to this list?