I am going to ask a simple question, for sure.
The pressure with respect to the altitude is given by this formula

Where
( from Wikipedia )
In addition to this, we have L=gCp where g=9.80665 m/s2 and Cp is the constant pressure specific heat =1007 J/(Kg K)
Understood and thence the above formula can be written in this simple way:
P=P0(1−αh)β
where
α=gCp T0≈3.3796⋅10−5 m−1
β=Cp MR≈3.5081971
On the other side we learnt from the elementary physics that the formula for the pressure is also given by
P=P0+rhogh
Where rho is the air density (1.23 kg/m3).
The question
First thing first: I assumed that
h=h1−h0
is that correct? I mean it's the difference between two heights (maybe from a table and the ground, just to say).
Since the two formulae seems quite different, I tried with a numerical example in calculating the pressure in two points, with a difference of height about 0.18 m and I got a really similar result.
Since the first formula is more technique, I think it's the correct formula but I would like to understand if one could pass from the first to the second or vice versa somehow.
Also I would like to know if there are cases in which I can use only the second formula or only the first formula!