We compare the thermodynamic behavior of a finite single nongrafted polymer near an attractive substrate with that of a polymer grafted to that substrate. After we recently found first-order-like signatures in the microcanonical entropy at the adsorption transition in the nongrafted case, and given the fact that many studies on polymer adsorption in the past have been performed for grafted polymers, the question arises, to what extent and in what way does grafting change the nature of the adsorption transition? This question is tackled here using a coarse-grained off-lattice polymer model and covers not only the adsorption transition but also all other transitions a single polymer near an attractive substrate of varying strengths undergoes. Because of the impact of grafting especially on the translational but also on the conformational entropy of desorbed chains, the adsorption transition is affected the strongest. Our results are obtained by a combined canonical and microcanonical analysis of parallel tempering Monte Carlo data.