We consider two recently proposed measures of non-Markovianity applied to a particular quantum process describing the dynamics of a driven qubit in a structured reservoir. The motivation for this study is twofold: on one hand, we study the differences and analogies of the non-Markovianity measures, and on the other hand, we investigate the effect of the driving force on the dissipative dynamics of the qubit. In particular we ask if the driving force introduces new channels for energy and/or information transfer between the system and the environment or if it amplifies existing ones. We show under which conditions the presence of the driving force slows down the inevitable loss of quantum properties of the qubit. © 2011 American Physical Society.