Non‐native trees may have significant impacts on the carbon sink capacity of forested lands. However, large‐scale patterns of the relative capacity of native and non‐native forests to uptake and store carbon remain poorly described in the literature, and this information is urgently needed to support management decisions. In this study, we analyzed 17,065 plots from the Spanish Forest Inventory (covering c. 30 years) to quantify carbon storage and sequestration of natural forests and plantations of native and non‐native trees under contrasting climate types, while controlling for the effects of environmental factors (forest structure, climate, soil, topography, and management). We found that forest origin (non‐native vs. native) highly influenced carbon storage and sequestration, but such effect was dependent on climate. Carbon storage was greater in non‐native than in native forests in both wet and dry climates. Non‐native forests also had greater carbon sequestration than native ones in the wet climate, due to higher carbon gains by tree growth. However, in the dry climate, native forests had greater carbon gains by tree ingrowth and lower carbon loss by tree mortality than non‐native ones. Furthermore, forest type (classified by the dominant species) and natural forests versus tree plantations were important determinants of carbon storage and sequestration. Native and non‐native Pinus spp. forests had low carbon storage, whereas non‐native Eucalyptus spp. forests and native Quercus spp., Fagus sylvatica, and Eurosiberian mixed forests (especially not planted ones) had high carbon storage. Carbon sequestration was greatest in Eucalyptus globulus, Quercus ilex, and Pinus pinaster forests. Overall, our findings suggest that the relative capacity of native and non‐native forests to uptake and store carbon depends on climate, and that the superiority of non‐native forests over native ones in terms of carbon sequestration declines as the abiotic filters become stronger (i.e., lower water availability and higher climate seasonality).