The absolute value of the inner product between two (pure) states $\lvert\psi\rangle$ and $\lvert\phi\rangle$, $\lvert\langle\psi\rvert\phi\rangle\rvert$, can be used to quantify the distance between the two states, and is commonly referred to as fidelity (though the fidelity is often defined as the square of $\lvert\langle\psi\rvert\phi\rangle\rvert$).
If the current state is $\lvert\phi\rangle$, and $\lvert\psi\rangle$ is a possible outcome of a measurement, then $\lvert\langle\psi\rvert\phi\rangle\rvert^2$ is the probability of getting the outcome $\lvert\psi\rangle$.
More generally, $\lvert\langle\psi\rvert\phi\rangle\rvert^2$ can be thought of as a measure of indistinguishability of the two states. This quantity will thus be important every time one wants to figure out what is the current state of the system, as a high fidelity implies that a lot of measurements are necessary to tell the two states apart.
An example of a quantum algorithm computing this quantity is the C-SWAP test, which I think was introduced in (Buhrman et al. 2001). One usage of this algorithm that comes to mind is given in (LLoyd et al. 2013), where they use it as a subroutine for their supervised cluster assignment algorithm (see end of pag. 3).
More generally, the inner product between two states is their overlap, which is such a fundamental quantity in quantum mechanics that it is hard to say how it can not be "useful" for any kind of protocol.