Examples would include things like
$$f(x, y) = \begin{cases} x^y & \text{ if } (x, y) \neq 0 \\ 0 & \text{ else} \end{cases}$$
versus
$$g(x, y) = \begin{cases} x^y & \text{ if } (x, y) \neq 0 \\ 1 & \text{ else} \end{cases}$$
both of which have their uses; or $n \mapsto H_n(X)$ versus $n \mapsto \tilde{H}_n(X)$ in algebraic topology.
How about the Liouville function, $$\lambda(n)=(-1)^{\Omega(n)},$$ versus the Möbius function, defined as $$\mu(n) = \begin{cases} (-1)^{\Omega(n)} && n \text{ is squarefree} \\ 0 && n \text{ is not squarefree} \end{cases}$$
($n$ is the product of $\Omega(n)$ primes.)
They differ on infinitely many points, but they are also the same on infinitely many points.