In this paper, we study the SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with stochastic network simulations to evaluate the impact of different weight distributions on epidemic thresholds and dynamics in general. For the SIR model, the basic reproductive ratio R0 is computed, and we show that (i) for both network models R0 is maximised if all weights are equal, and (ii) when the two models are 'equally-matched', the networks with a random weight distribution give rise to a higher R0 value. The models with different weight distributions are also used to explore the agreement between the pairwise and simulation models for different parameter combinations.