Fluids are complicated systems described by non-linear differential equations that can't be reasonably treated in a full generality (certainly not analytically). Just consider the kinds of waves that propagate in the sea -- deep or shallow water, solitons, tsunami and many others (this is not to say that these are sound waves; but as an illustration of complicated wave behavior it should suffice). So, to proceed one often employs some approximation.
By far the most popular one (with many applications) rests on the linearization of the problem around an equilibrium solution where one replaces complicated non-linear equations with second-order wave equation that describes propagation of the perturbation in the system. Now, this places some consistency conditions on how big those perturbation can be so that higher-order effects can be ignored. Depending on the precise form of the equations, this might require that the temperature or some other parameter stays constant (otherwise the induced heat transport effects might destroy the linearization, for example)
Roughly, isothermal processes are slow (so that there is enough time for the transfer of the heat with the environment which keeps the temperature constant) whereas in the adiabatic case the wave propagates so fast that the environment can't catch up with it and so no heat is exchanged (but the temperature can change). I think for the most familiar types of materials where the speed of sound is quite big one uses the adiabatic approximation (certainly for the propagation of sound in the air). I guess for less standard materials (as encountered in astrophysics) you might need isothermal approximation too but it's hard to say more than this without knowing what system you have in mind precisely.