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

I don't really see the difference, because in both case we need to determine y and the values of the constants. The only difference is that we give the value of y and y' in the former and the value of either 2 y or 2 y' in the latter.

I solve both problems the same way. I don't really understand the theory, I guess.

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.

For a simple example (second order ODE), an initial value problem would say $y(a)=p$, $y'(a)=q$.

A boundary value problem would specify $y(a)=p$, $y(b)=q$.

Initial Value Problems:

In initial value problems, we are given the value of function $y(x)$ and its derivative $y'(x)$ at the same point ( initial point ) sy at $x = 0$ i.e $y(0)= xi1$ and $y'(0)= x_2$.

Boundary Value Problems:

In boundary value problem, we are given the value of function $y(x)$ at two different points, i.e $y(a)= x_1$ and $y(b)= x_2$.

Initial Value Problems:

Initial value problem does not require to specify the value at boundaries, instead it needs the value during initial condition. This usually apply for dynamic system that is changing over time as in Physics. An example, to solve a particle position under differential equation, we need the initial position and also initial velocity. Without these initial values, we cannot determine the final position from the equation.

Boundary Value Problems: In contrast, boundary value problems not necessarily used for dynamic system. Instead, it is very useful for a system that has space boundary. An example would be shape from shading problem in computer vision. To determine surface gradient from the PDE, one should impose boundary values on the region of interest.

Initial value problem will be given initial conditions. But the boundary value problem contains boundary conditions like y(x1) and y(x2).