The solar wind is a highly supersonic stream of ionized gas which creates a large bubble of plasma around the Sun. This bubble, termed the heliosphere, is stretched like a raindrop by the supersonic flow of partially ionized interstellar gas. In this paper we present a detailed comparison between the kinetic Monte Carlo and multiple neutral fluid approaches to modeling hydrogen atoms of interstellar and heliospheric origins within an axisymmetric heliosphere, coupled through charge exchange to the ionized plasma. We choose a set of parameters and boundary conditions which allow a further comparison with existing independently obtained results. Our results show a similar overall heliospheric structure, with a heliopause in approximately the same place, for both the kinetic and fluid models. The termination shock, on the other hand, is moved about 5 AU inward and the bow shock is weakened and moved about 8 AU farther out in the kinetic model. More critically, the hydrogen wall amplitude is about 50% above the interstellar value in the fluid case but only 25% in the kinetic case, and the level of hydrogen inside the termination shock is about 20% higher in the kinetic model. The velocity distributions at various locations along the symmetry axis show that while the kinetic model predicts a smooth distribution, the total fluid distribution is more disjointly structured. We also find that neutral atoms produced in the inner heliosheath have a "half-Maxwellian" distribution outside the heliopause, something which cannot be realized within the fluid model.