Perform the matrix addition on the left side of the equation.

$$\begin{bmatrix} -2 & x & 3 \\ z+3 & 4 & -y \end{bmatrix} + \begin{bmatrix} 2 & 1-x & -2 \\ -3 & 4+x & y \end{bmatrix}$$

Add the corresponding elements:

$$\begin{bmatrix} -2+2 & x+(1-x) & 3+(-2) \\ (z+3)+(-3) & 4+(4+x) & -y+y \end{bmatrix}$$

Simplify the elements:

$$\begin{bmatrix} 0 & 1 & 1 \\ z & 8+x & 0 \end{bmatrix}$$

From the element in row 1, column 1:

$$0 = 0$$

From the element in row 1, column 2:

$$1 = 1$$

From the element in row 1, column 3:

$$1 = 1$$

From the element in row 2, column 1:

$$z = 2$$

From the element in row 2, column 2:

$$8+x = 8+x$$

From the element in row 2, column 3:

$$0 = 4$$

The last equation, $0=4$, is a contradiction. This means that there are no values of $x$, $y$, and $z$ that can satisfy all the conditions of the given matrix equation simultaneously.

Therefore, there is no solution for $x$, $y$, and $z$.

The final answer is $\boxed{\text{No solution exists for } x, y, \text{ and } z}$.