Maximum entropy principle does not seem to distinguish between the use of Tsallis and Renyi entropies as either of them may be used to derive similar kind of q-exponential distributions. In this paper, we address the question whether the Renyi entropy is equally suitable to describe those systems with q-exponential behaviour, where the use of the Tsallis entropy is relevant. We discuss a specific class of dynamical systems, namely, one-dimensional dissipative maps at chaos threshold and make our study from two aspects: (i) power-law sensitivity to the initial conditions and the rate of entropy increase, (ii) generalized bit cumulants. We present evidence that the Tsallis entropy is more appropriate entropic form for such studies as compared to Renyi form. (C) 2003 Elsevier B.V. All rights reserved.