There's no difference from a mathematical point of view (As far as I recall).
In physical systems depending on time, it's usually customary to state to values of the function at time t=0 (or t=t0).
When dealing with a time-independent problem, you want to give the values at some point x=x0.
For the first, initial value is used, for the second, the term 'boundary condition' is more customary.
Boundary conditions are used often in PDE's, such as the Laplace equation for f(x,y,z), since there is no time dependence.
In QM, we can state \Psi(t_0,x) which is a initial value for the Schrodinger equation. Or we can state for example \Psi_(t,0)=0, which is a boundary condition.