The maximum clique and minimum vertex cover problems are among Karp's 21 NP-complete problems, and have numerous applications: in combinatorial auctions, for computing phylogenetic trees, to predict the structure of proteins, to analyse social networks, and so forth. Currently, the best complete methods are branch & bound algorithms and rely largely on graph colouring to compute a bound. We introduce a new approach based on SAT and on the "Conflict-Driven Clause Learning" (CDCL) algorithm. We propose an efficient implementation of Babel's bound and pruning rule, as well as a novel dominance rule. Moreover, we show how to compute concise explanations for this inference. Our experimental results show that this approach is competitive and often outperforms the state of the art for finding cliques of maximum weight.