I think it's a difficult question - some human concepts (particularly emotions) are based on our experience, which is not necessarily identical to that of a different species. I work in pain research, where the distinction between nociception (i.e. sensory detection of something harmful) and pain (i.e. cognitive and emotional response to nociceptive stimuli) is quite important when trying to understand similarities and differences between ourselves and other species - nociception is generally the preferred term.

I think terms like behaviour and personality can be used for both in the right context, but I have issues applying words like 'anxiety' and 'depression' to animals as these are emotional concepts reliant on a level of cognitive sophistication we have no evidence for in most other organisms - it's much more informative to try and describe any behaviours observed in a more objective way, without making assumptions about emotional/cognitive input.

It's tricky!

Yes, but the other way round is also true. We can learn behaviors, characteristics and features from non-human species that may apply to humans. One good example is "swarm intelligence", that has been studied originally in herds, bird flocks, school fish, social insects - and then, later on, it's been transposed on to humans, too.

True, although isn't most true swarm intelligence governed by non-verbal communication e.g. pheromones? Don't know I'd agree that the term can be applied directly to human behaviours - I mean, it can be used for artificial modelling but I think it's always important to think of the context in which you are applying the term. There is no simple answer I'm afraid!

I'm no expert in this area but I'd like to share my opinion. I agree that anthropomorphisms add biases that impact our efforts to objectively understand the traits of other species. In my attempt to perceive animal behaviour outside of my human mind, I have classified responses to stimuli (behaviour) as being an action that either positively or negatively impacts reproductive potential, or the recognition thereof. Personalities are just our way of identifying where someone sits in a social network/hierarchy (e.g. dominant or submissive) but they all impact reproductive potential. This creates an objective identification of a behaviour/personality that applies to all species on a very general scale. I don't know if this is sound, completely misguided or "behind the 8-ball" because it's not my field and I'm just thinking out loud.

Some animals do in fact have a personality. You are one of them. Even hamsters have personalities. I disagree with anthropomorphism; that's the mistaken direction Darwin's son went in. 

I think it should be BOTH and not VS. For instance, Physics (a non-human science) has informed Economics (a human science). And Biology (a nonhuman, animal science) has informed Physics (a nonhuman, non animal science).  And if we try to make the connections through holism we can infer a bidirectional influence between non-human and human studies. Since if we try to juggle the two opposing perspectives it will eventually give us a common denominator (if we know the two fairly well: non-human & human)- dialectical resolution?. For instance, I like the analogy of "swarm intelligence", highlighted by Carlos Eduardo Maldonado, being transposed on to human behavior. Even though the idea of human pheromones is controversial, it nonetheless points to a more fundamental, underlying, common denominator- the force of influence (e.g. leadership, pecking order and the like). So the analogue to pheromones may take the form of instinctive behavior and non-rational tendencies given certain cues-which behavioral economists are beginning to unravel, and some have referred to as "fast-thinking" vs the "slow thinking" of the homo economicus. In this sense I agree with you that employing two different terms is superfluous except for identification purposes. Isn't this why scientist  tend to boil everything down to the maths- a system of neutral, symbols?

http://www.amazon.com/Thinking-Fast-Slow-Daniel-Kahneman/dp/0374533555