First-order (linear) error propagation approximates f near the mean with a Taylor expansion, giving a closed-form output variance from the partial derivatives:
σ_z² = (∂f/∂x)²·σ_x² + (∂f/∂y)²·σ_y²
Monte Carlo instead draws thousands of (x,y) pairs from Gaussian distributions with the chosen means and standard deviations, evaluates z = f(x,y) for each pair directly, and measures the empirical spread. For linear formulas (sum, difference) the two methods agree almost exactly at any σ. For nonlinear formulas (product, square, hypotenuse) push σ_x or σ_y up and watch the histogram skew away from the red analytical curve — that gap is exactly what "relative difference" reports.