Motivated by recent experiments on chemically synthesized magnetic molecular chains, we investigate the lowest-lying energy band of short spin-s antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By numerical diagonalization we find that the Landé pattern in the energy levels, i.e., E(S)â̂S(S+1) for total spin S, known from, e.g., ring-shaped nanomagnets, can be recovered in odd-membered chains, while strong deviations are found for the lowest excitations in chains with an even number of sites. This particular even-odd effect in the short Heisenberg chains cannot be explained by simple effective Hamiltonians and symmetry arguments. We go beyond these approaches, taking into account quantum fluctuations by means of a path-integral description and the valence bond basis, but the resulting quantum edge-spin picture which is known to work well for long chains does not agree with the numerical results for short chains and cannot explain the even-odd effect. Instead, by analyzing also the classical chain model, we show that spatial fluctuations dominate the physical behavior in short chains, with length Nas, for any spin s. Such short chains are found to display a unique behavior, which is not related to the thermodynamic limit and cannot be described well by theories developed for this regime.