What's the difference between an initial value problem and a boundary value problem?

An initial value problem is how to aim my gun. A boundary value problem is how to aim my gun so that the bullet hits the target.

Qualitatively the methods of solution are sometimes different, because Taylor series approximate a function at a single point, i.e. at 0.

Here is an example:
For an ordinary 2nd order linear differential equation
y" + f(t) y' + g(t)y = 0,

- the initial value problem is to find the partial integral (solution) of the equation which satisfies the initial conditions
y(t0) = a, y'(t0) = b (t0, a and b are arbitrary constants);

- the boundary value problem is to find the partial integral which satisfies the boundary conditions
y(t0) = c, y(t1) = d (t0, t1, c and d are arbitrary constants, t0 < t1).

In other words: the initial value problem involves conditions in one point (the initial point of an interval), the boundary value problem involves conditions in various points (boundary points of the interval).