The key similarity between 4e hit points and pre-4e hit points is that any hit that does not reduce a character to 0 hit points or less is a non-threatening wound that does not hamper the character's ability to fight.
A high-level character who is low on hit points will look pretty much the same regardless of edition: covered with nicks, scratches and bruises, out of breath, and probably low on luck as well. The process of getting him to that point will also look pretty much the same, regardless of edition: due to skill, luck, and other factors, attacks that would have killed a normal man have been evaded and/or turned into minor injuries. In addition, the next time he gets hit, the same thing will happen regardless of edition: he will sustain a life-threatening wound that will kill him if he is unlucky or if he does not get help.
The first key difference between 4e hit points and pre-4e hit points is that the only way to recover hit points quickly pre-4e was to use magic. 4e hit points are more like vitality points in that they can be recovered quickly without magical assistance, e.g. being affected by a warlord's inspiring word, spending healing surges during a short rest, or recovering all healing surges after an extended rest. Pre-4e, a character's ability to convert a serious wound into a minor injury can only be regained slowly without the use of magic. In 4e, a character can replenish his skill, luck and the other "intangible" aspects of hit points more quickly without magical assistance. A 4e character who has been brought down to low hit points and then recovers all his hit points after a rest is still covered in nicks, scratches and bruises, but his ability to convert future serious wounds into minor injuries has been regained.
The second key difference between 4e hit points and pre-4e hit points happens when a character has been brought to 0 hit points or less. It is actually a subset of the first key difference. Pre-4e, a character who has been brought to 0 hit points or less was either automatically dead or dying (in 3e, or if you used the dying optional rule pre-3e). If magical healing was available, a dying character that was restored to 1 hp or more continued to function normally. Whatever serious, life-threatening injury he sustained was either healed or converted to a minor, non-threatening wound that does not hamper his ability to fight. If magical healing was not available, bed rest could achieve the same effect, but more slowly. Even so, for 3.5e characters of 10th level or higher, eight hours of rest was all that was necessary for them to go from almost dead (-9 hp) to functioning normally (1 hp or more). And at this point, we are back to the first key difference between 4e hit points and pre-4e hit points.
So, what happens when a 4e character has been brought to 0 hit points or less and then gets his hit points restored in a non-magical fashion? Somehow, that serious wound that he sustained is no longer life-threatening and no longer hampers his ability to fight. During a fight, this could be explained by a rush of adrenaline or being so inspired by an ally that the character functions normally despite his wounds. After a short or extended rest, it could be explained by treating and binding the character's injuries so that he functions normally despite his wounds. Alternatively, the character could be made of such stern stuff that after a short period of gritting his teeth, he just functions normally despite his wounds. Non-magical healing doesn't make a wound go away. It just allows a character to function normally despite his wounds.
Now, not liking that non-magical healing can allow a character to function normally despite his wounds is a valid complaint, and there are a number of possible sub-systems that can address it, e.g. lingering wounds (using the disease track mechanic), temporary reduction in the number of healing surges per day, temporary limits to the character's maximum hit points, etc.
However, I do believe that 4e hit points and pre-4e hit points handle damage to the character in pretty much the same way. The two key changes that 4e has made to hit points are that the "intangible" aspect of hit points that allow a character to turn a serious wound into a minor injury are recovered more quickly in 4e, and that 4e characters who have not received magical healing are still able to function normally despite their wounds.