Let \(\varphi_j\), \(j=1,2,3,4\), denote holomorphic self-maps of the unit disk \(\mathbb{D}\) of \(\mathbb{C}\). Set \[T_{kj} = C_{\varphi_k} - C_{\varphi_j},\] where \(C_{\varphi_k}\) denotes the composition operator defined on the space of holomorphic functions in \(\mathbb{D}\) as \(C_{\varphi_k} f = f\circ\varphi_k\). Set \[ T = T_{12} - T_{34} = T_{13} - T_{24}. \] Thus, \(T\) is a double difference of composition operators. Let \(\rho\) denote the pseudo-hyperbolic distance on \(\mathbb{D}\). Put \(\rho_{kj}(z) = \rho(\varphi_k(z), \varphi_j(z))\), \(z\in \mathbb{D}\), and \[ M_{kj}(z) = \left( \frac{1-|z|}{1-|\varphi_k(z)|} + \frac{1-|z|}{1-|\varphi_j(z)|} \right)\rho_{kj}(z), \quad k,j=1,2,3,4. \] For \(\alpha>-1\) and \(0< p<\infty\), the authors prove that the double difference \(T\) is compact on the weighted Bergman space \(A^p_\alpha(\mathbb{D})\) if and only if \[ \lim_{z\to 1-} [M_{12}(z) + M_{34}(z)][M_{13}(z) + M_{24}(z)] =0. \] As an application, they construct an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, double difference cancellation is not possible in a certain local sense.