The "Divide and Concur'' (DC) algorithm, recently introduced by Gravel and Elser, can be considered a competitor to the belief propagation (BP) algorithm, in that both algorithms can be applied to a wide variety of constraint satisfaction, optimization, and probabilistic inference problems. We show that DC can be interpreted as a message-passing algorithm on a constraint graph, which helps make the comparison with BP more clear. The "difference-map'' dynamics of the DC algorithm enables it to avoid "traps'' which may be related to the "trapping sets'' or "pseudo-codewords'' that plague BP decoders of low-density parity check (LDPC) codes in the error-floor regime. We investigate two decoders for low-density parity-check (LDPC) codes based on these ideas. The first decoder is based directly on DC, while the second decoder borrows the important "difference-map'' concept from the DC algorithm and translates it into a BP-like decoder. We show that this "difference-map belief propagation'' (DMBP) decoder has dramatically improved error-floor performance compared to standard BP decoders, while maintaining a similar computational complexity. We present simulation results for LDPC codes on the additive white Gaussian noise and binary symmetric channels, comparing DC and DMBP decoders with other decoders based on BP, linear programming, and mixed-integer linear programming.