The basic concept of freeze drying is, to freeze the perishable product, then using some type of vacuum, lower the pressure in a controlled fashion, to allow the water, in a solid state at this point, to convert directly to a gas, without going through the intermediate liquid state. I don't exactly understand the process of sublimation. But the idea is that by using this process in a controlled way, you can remove the moisture easily in a controlled fashion, easily preserving it, and leaving almost everything else. But of course, most living things are primarily water, and are definitely water dependent, so removing water cannot leave any organic product unchanged.
As for the health benefits and which food is more healthful for a pet, it really depends on the food itself being freeze dried. I am already on record saying that most freeze dried insects are a shell of their former selves, both literally and figuratively. From what I have seen, plant materials seem to hold more nutrition than most protein based (animal) materials. I am sure there are exceptions, I haven't studied all of them. It is mostly dependent on how easily the rest of the chemicals in the original product lend themselves to sublimation, and how the removal of all or part of those products in addition to the water affect what is left in the remaining product, nutritionally and otherwise.
I'll now leave this question to the bio-chemist, that should hopefully wonder by anytime.