Chemical reaction networks (CRNs) are used to model a variety of chemical and biological processes. A CRN may be viewed as a directed graph, whose nodes are chemical complexes and whose edges are chemical reactions. Wegscheider proposed, in 1901, the principle of detailed balance for chemical kinetics resulting from microscopic reversibility. In the deterministic setting of a dynamical system, the principle of detailed balance manifests as a set of conditions on the rate constants arising from the graphical structure of the CRN. When modeled as a Markov chain, a different graphical structure is associated with a CRN, one whose nodes are population numbers of the chemical species, and whose edges are transitions with non-zero probabilities. We show that the two notions -- Deterministic Detailed Balance (DDB) and Stochastic Detailed Balance (SDB) -- are intimately related, but not equivalent. In particular, DDB implies SDB, but the converse is not true. However, achieving SDB without DDB requires stringent conditions on the rate constants, that are rarely realized in practice. An important exception to this is a birth and death process, for which SDB always holds but DDB does not hold in general.