A key feature of biological systems is the emergence of higher-order structures from interacting units, such as the development of tissues from individual cells and the elaborate divisions of labor in insect societies. Little is known, however, of how evolutionary competition among individuals affects biological organization. Here we explore this link in bacterial biofilms, concrete systems that are well known for higher-order structures. We present a mechanistic model of cell growth at a surface, and we show that tension between growth and competition for nutrients can explain how empirically observed patterns emerge in biofilms. We then apply our model to evolutionary simulations and observe that the maintenance of patterns requires cooperation between cells. Specifically, when different genotypes meet and compete, natural selection favors energetically costly spreading strategies, like polymer secretion, that simultaneously reduce productivity and disrupt the spatial patterns. Our theory provides a formal link between higher-level patterning and the potential for evolutionary conflict by showing that both can arise from a single set of scale-dependent processes. Moreover, and contrary to previous theory, our analysis predicts an antagonistic relationship between evolutionary conflict and pattern formation: conflict drives disorder.