It's difficult to say without having any first-hand knowledge of such a phenomenon, though I would imagine that a good model for a person living to the age of 600 would be to take the central differences between a child of 6 and a grown person of age 60. Why 60? Because this is a safe age to extrapolate behavior, after which, behavior might begin to suffer from the onset of old age. Since they will have the body of a 30-year-old, I assume that diseases such as alzheimer will not have an effect.
So what are the central differences between a 6-year-old and a 60-year-old? This in of itself is a difficult question to answer because it mostly depends on the individual, however I think we can agree that generally, you see the following traits in 6-year-olds that you wouldn't see in 60-year-olds:
- Egocentric world view
And so logically the type of traits you'd typically see in 60-year-olds (and again, not always) would be Maturity, Accepting world view, Knowledgeable, Rationality, Patience. Therefore presumably a 600-year-old is just a more extreme version of this.
Extreme maturity - They likely wouldn't appreciate humor as we appreciate humor, and even then, they would likely find little funny, save for very sophisticated and subtle humor which might easily go over our heads. For the most part, they'd be like stereotypical tight-lipped librarians that want nothing to do with the silly things you do.
Extreme accepting world view - Little is likely to surprise them, since they've seen everything before. The ideas that they've had over the years has solidified into granite and therefore they realize that they aren't going to change ideas. And it's not without a certain irony that this also means that they've made up their minds regarding their approach to such contrasting ideas. You aren't likely going to be able to trick one into arguing with you, but if you could and assuming you were listening, it is far more likely they'd tell you something to put your own ideas into question. Their general approach to differing world views is likely going to be apathy.
Extreme knowledge - There will be little that they do not know. Although they may not have knowledge on par with an expert in the field, they will certainly have enough base knowledge to ask a question that may stumble an expert in most fields. The type of knowledge known for the fields that interest them is unfathomable, though suffice to say that they will likely learn neighboring fields at the level of expert simply because the two fields have some common ground. In other words, a 600-year-old biologist would likely be able to tell you anything you'd want to know in the areas of chemistry or zoology, at the extreme of being considered an expert in both fields, as well as quote the first chapter of Dante's Inferno by heart just because it is a hobby of his.
Extreme rationality - A 600-year-old man would be able to argue circles around you, thinking several steps ahead. This is not necessarily because he is smarter than you, but simply because he's had a lot of time to practice. Suffice to say that after you explain the rules of chess to such a man, he'd likely not lose a single match and by the third game, he'd have figured out how to beat you in 10 moves or less playing off your own strategy. This isn't to say that such a person couldn't be wrong, but his beliefs would be deeply rooted in fact that you couldn't prove him wrong without first showing that one of these facts is fundamentally wrong as well.
Extreme patience - You would not likely see a 600-year-old man show a temper. Combined with extreme rationality and you'd feel like you were talking with a living version of Spock. In fact it would be difficult to evoke a strong sense of emotion of any kind. These types of emotions would be likely limited to only friends and loved ones and only in specific occasions.
Could a 600-year-old man be immature, egocentric, innocent, irrational, impatient? Yes, absolutely. This is merely a generalization, though you likely wouldn't see 600-year-old individuals if they did show these traits, because like children, they would "grow out of it" more likely than not.