In this article, we introduce a new and simple data structure, the prefix table under Hamming distance, and present two algorithms to compute it efficiently: one asymptotically fast; the other very fast on average and in practice. Because the latter approach avoids the computation of global data structures, such as the suffix array and the longest common prefix array, it yields algorithms much faster in practice than existing methods. We show how this data structure can be used to solve two string problems of interest: (a) approximate string matching under Hamming distance; and (b) longest approximate overlap under Hamming distance. Analogously, we introduce the prefix table under edit distance, and present an efficient algorithm for its computation. In the process, we also define the border array under both distance measures, and provide an algorithm for conversion between prefix tables and border arrays.