Measurements of inclusive and differential production cross-sections of a topquark-top-antiquark pair in association with a W boson (t (t) over bar W) are presented. They are performed by targeting final states with two same-sign or three isolated leptons (electrons or muons) and are based on root s = 13TeV proton-proton collision data with an integrated luminosity of 140 fb(-1), recorded from 2015 to 2018 with the ATLAS detector at the Large Hadron Collider. The inclusive t (t) over bar W production cross-section is measured to be 880 +/- 80 fb, compared to a reference theoretical prediction of 745 +/- 50 (scale) +/- 13 (2-loop approx.) +/- 19 (PDF, alpha(s)) fb. Differential cross-section measurements characterise this process in detail for the first time. Several particle-level observables are compared with a variety of theoretical predictions, which generally agree well with the normalised differential cross-section results. Additionally, the relative charge asymmetry of t (t) over bar W+ and t (t) over bar W- is measured inclusively to be Arel C = 0.33 +/- 0.05, in very good agreement with the theoretical prediction of 0.322 +/- 0.003 (scale) +/- 0.007 (PDF), as well as differentially.