I have seen two different forms of Schrödinger equation: $$i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}=\hat{H}|\psi(t)\rangle$$ and $$i\hbar\frac{d|\psi(t)\rangle}{d t}=\hat{H}|\psi(t)\rangle.$$
Are these two equations equivalent? If not, in what situations are each equation used?
There is no difference. Strictly speaking the second one is correct, because the state vector is a function only of time, but physicists aren't always very careful about distinguishing partial vs single-variable derivatives. The partial derivative might have stuck because as an analogy to the wavefunction $\psi(x,t)$, for which a partial derivative is appropriate.
