What I liked about this paper initially are the titles of the responses, which include "Insects cannot tell us anything about subjective experience or the origin of consciousness" and "Consciousness explained or consciousness redefined?" Actually, that second one is not so appealing: very unfortunately, the most interesting philosophical discussion on consciousness has spent an awful lot of time focusing on definition, and it seems that there isn't much more to be said. So cries of misdefinition aren't very satisfying unless the chosen definition is really outré, which this one doesn't appear to be.
In fact, the authors dive right in and attack the central issue, which is subjective experience. They argue that insects possess analogous structures to areas of the human midbrain which integrate sensory experiences, that the midbrain in humans is responsible for subjective experience, and that therefore these same structures in insects, having the same function, thus may give rise to subjective experience.
Obviously, subjective experience isn't purely a result of sensory integration, the classical objection being that if it were, that would bestow consciousness on such unlikely things as an automatic swimming pool pump or a thermostat. This is where the paper gets a bit lost, in my opinion. Nematodes, with their central nervous systems, are capable of sensory integration, but are denied subjective experience in this paper because it's not obvious that they are capable of using that integrated information in an "egocentric" way -- for example, to find food when hungry. Selective attention and the ability to create memories, both of which seem somewhat related to egocentric employment of sensory integration areas, are also referenced, but not given much discussion: "because insects clearly have a capacity for selective attention we may safely sidestep this debate for now".
A single paper could hardly be expected to completely address the issue of consciousness, and it's totally reasonable (in fact, it's a good thing) that this one doesn't attempt to. It makes a good case for the localisation of human subjective experience in the midbrain and for analogous structures in insects. It does quite well at avoiding what I've come to think of as "consciousness chauvinism", in which humans are obviously conscious but nonhuman display of what looks like consciousness is obviously just the stimulus-and-response patterns of a simple biological machine. It also makes the overall point that subjective experience is a useful trait for all sorts of creatures, not just humans. So perhaps the question shouldn't be framed as "do insects have subjective experience?" Instead, maybe we should be asking: if insects don't have it, why not?