I'd probably identify three levels (or, at least, mark three areas on a continuum) but for different reasons. There's the class of organisms so far removed from us that analogies are difficult to make even if they do exhibit reactions to noxious stimuli (single cell organisms, worms, insects, etc). Then there's the continuum of animals from, say, simple vertebrates to mammals to primates, where their form of life is increasingly similar to our own, and it becomes much easier to identify when they're in pain. However, all such pain-identifications are attenuated and the final class is language-using beings such as ourselves, who can exhibit a far more nuanced range of suffering because of our developed linguistic skill. I think for most people the cut off point will be somewhere along the continuum of the second class and is somewhat arbitrary.
I do think when trying to identify the range of application of a psychological concept like pain it has to be relative to the concept originator (human beings). Pain is not simply a mental state or a behaviour, it's part of a complex conceptual network, partly constituted by behaviour, sensations, etc, but dependent on the antecedent applicability of a range of other concepts. Pain applied to animals is always in an attenuated sense. The more distant these animals in their lifestyles compared to our own, the less applicable the concept, since we have to find home for an entire nexus of concepts relating to pain in order to apply the concept of pain. It's an isomorphism. You can fill in gaps, ignore dissimilarities, etc, but eventually there's enough distance that you just can't fit one to the other at all.
I ended up reading this article about animal suffering by this Christian apologist called William Craig. Forgive the source, please.
He continues the argument here.
How decent do you think this argument is? I don't know where to look to evaluate the core claim, as I know very little neuroscience myself. I'm quite concerned about animal suffering, and choose to be vegetarian largely on the basis of that concern. How much should my decision on that be affected by this argument?
EDIT: David_Gerard wins by doing the basic Google search that I neglected. It seems that the argument is flawed. Particularly, animals apart from primates have pre-frontal cortexes.