For the most part they are identical, the only identifiable difference is that military spec tube webbing has a ribbed weave, while climbing spec has a smooth weave. Having a smooth weave obviously makes tube webbing better for tying and most importantly, untying knots; it also makes it a lot easier to pull through your carabiners, especially when stacked. Climbing spec is also slightly stronger – ~19kN vs 17.6kN.
Other than that, there's no real difference, the primary reason why you would use one over the other would be availability. Military spec webbing is more common in stores in some regions, and comes in a larger variety of colours (like digital camo).
Military Spec: (Ribbed)

Climbing Spec: (Smooth)
