The relative importance of abiotic and biotic factors in shaping forest biomass stocks and fluxes remains a controversial issue. Here, using data gathered from 39 1 ha plots located in flooded and terra firme mature tropical lowland forests of the Amazon and Orinoquia regions of Colombia, we evaluated the importance of climate, soil fertility, and flooding, as well as tree taxonomic/phylogenetic diversity and forest structural properties, in determining the aboveground biomass stocks (AGB; Mg ha−1) and aboveground woody productivity (AWP; Mg ha−1 year−1). Using information‐theoretic multimodel inference and variance partitioning we found that forest structural features such as the number of trees with diameter at breast height ≥ 70 cm, and wood density, are the main drivers of variation in AGB. However, taxonomic diversity also contributes to AGB because it is associated with more large trees in these forests. In contrast, the key drivers of AWP in these forests were soil P and Mg concentrations, with no significant effects of diversity indices. These findings emphasize the need to include major soil cations other than N and P (e.g. Mg) in experimental studies to improve our understanding about the extent to which soil fertility can modulate increases in forest AWP due to climate change. Terra firme forests had higher AGB stocks than flooded forests, but both had similar AWP; and we found similar results for the drivers of AGB and AWP between flooded and terra firme forests. Our results provide limited evidence for strong effects of plant diversity on AGB or AWP. Therefore, we call for caution on generalizations of nature‐based initiatives aiming to preserve diversity based on maximizing carbon stocks and productivity, due to the complex nature of the processes controlling carbon accumulation and carbon fluxes in tropical forests.