Insufficiency of point-like approximation
Movement of a human body, such as marching or climbing the stairs, is not well approximated by a point-like object, which is a usual approximation in basic mechanics courses. While this could be an excellent example to demonstrate the importance and the limitations of the basic assumptions, most textbooks and teachers choose to approximate human by a point in order to focus more mundane applications of basic formulas.
Thermodynamic point of view
As mentioned in the answer by @Anonymous, the energy spent by the human body goes not only into changing the gravitational energy, but in making the body parts move. Notably, even the movement that does not change gravitational energy cost some energy, which is dissipated as heat into the muscles. Thus, if we look closer at the dynamics of human body during walking up the stairs and climbing the ladder, these two are clearly different:
- When walking up the stairs, one moves a foot to the next stair, first lifting it almost vertically and then displacing it horizontally. Then one shifts the weight horizontally to this foot and lifts this weight vertically.
- When climbing the staircase, one typically holds with one's hands to prevent oneslef from flipping over, and performs with legs the cycle of movements similar to the one described above, perhaps helping with one's hands when pulling the body up.
One can look at this from the point of view of thermodynamics, as two different heat engine cycles, which are likely to have different efficiency, i.e., requiring different energy to perform the same useful work (i.e., changing the mechanical energy by $mgh$). One could perhaps argue that walking up the stairs is more economical, as one does not have to spend any energy while not moving, which is not the case when climbing the ladder. Adopting an evolutionary viewpoint one could even argue that this is the reason why stairs (rather than ladders) are the preferred method of climbing up.
Heat engine cycle
With a bit of reflection the heat engine picture can be worked out in more details. For simplicity I will present here a simple model of walking, but it is readily generalizable to climbing stairs. We consider as the rest state the pisition where the two legs are parallel ti each other. As the state variables we consider the angle $\theta$ between the two legs (in the reste position $\theta=0$), and the position of the center-of-mass of the body, $h$. The cycle of walking then consists of three stages:
- Lifting one leg. This is a diagonal in $\theta-h$ plot, as the legs open and the center-of-mass moves upwards. An amount of energy $Q_1$, necessary for lifting the center-of-mass, is supplied by metabolic reactions in the muscles.
- Free fall the body falls to the lifted leg, the center-of-mass goes down, but the angle between the legs does not change, i.e. this part of the cycle is a straight line in the state coordinatrs. No metabolic energy is spent, i.e. the process is adiabatic.
- Closing legs, returning the body to the initial position. This is again a line in our $\theta-h$ plot. Amount of metabolic energy is again spent.
The cycle forms a triangle in $\theta-h$ coordinates, the area of which is the work done for movibg one step.
One can now choose a mechanistic model of human legs, e.g., two bars connected by a joint, to relate the two variables in the plot and estimate the energy spent on making one step.
Note that naive simple freshman year mechanics predicts that this work is zero!