Two items that happens physically to a human body that is never really addressed when having a character in a story having a greatly increased lifespan: changes in brain activity and skin elasticity. There are many stories that have humans with increased lifespan through some method that "maintains the body at its peak condition"... I am NOT addressing those stories as these effects I am talking about are "ageing" effects and any method that maintains peak condition is by definition removing ageing effects. I am talking about the stories with longevity through other means like a curse/spell, curing illnesses/diseases, organ replacement when failing through cloning, etc. These things would all extend the human lifespan without addressing the brain or skin elasticity.
1)changes in brain activity: the longer a body is alive, the more changes a person's brain goes through. The area of the brain involved in learning shrinks slowly over large periods of time meaning that it takes longer and longer to learn new skills the older a person is. Communication between neurons becomes less effective over time meaning that it takes longer to recall specific words, memories, etc. Other similar changes occur that overall would have a degenerative effect on brain function. These changes are a natural side-effect of long life and any stories that deal with an increase in human longevity through any other means than a method that "keeps the body at peak human condition" (which would obviously include the brain) would have to have developed a means to deal with those as well.
2)skin elasticity: human bodies reach their "peak condition" somewhere between 20-25 years old. From that "peak" point onwards, the human body is in a slow decline. This includes the skin. Over time the skin loses its elasticity and you start seeing the effects of gravity more.. ie wrinkles. But it is not ONLY wrinkles that showcase the loss of skin elasticity. It is also things like the skin being much easier to bruise or to tear...for skin that has lost a lot of elasticity (ie is much older) it is possible to bruise or even tear the skin through such simple things as a firm handshake, a light clap on the back, going through a doorway and bumping their shoulder on the door frame. Also, over a very long period of time, the loss of skin elasticity would be seen in the shape and sizes of a person's ears. The ear would seem to lengthen over time (due to the combination of gravity and loss of skin elasticity on the protruding skin) at a rate of roughly 1cm every 40-50 years... regular senior aged people it would not be very noticeable... but that would be a full added inch to ear length over about 120 years. Although there are methods to combat skin elasticity (face lifts for example), there is only so much that can be done and eventually the skin will show the effects of ageing.
So just remember when writing the characters in your stories: if dealing with trying to show advanced age/longevity in any manner NOT attributable to "maintaining peak condition", instead of just saying something like so-and-so's face was "a mass of wrinkles" you can show their advanced age by saying something like "their skin was almost translucent - thinned out by their age, and deep wrinkles lined their face. Their ears seemed longer than normal, the lobes drooping toward the ground. Their words were slow and measured, each seemingly chosen with care instead of falling without thought from their mouth. Their movements were equally slow and careful as they settled into their chairs at the table." This type of description would highlight the longevity of the character a LOT more than the more common "mass of wrinkles" or "wrinkle-lined face" that I see in a lot of stories.
As always this is just my thoughts/opinion.