An Inconsistency in Numerical Approximation

It is simply an issue of accuracy of approximation. Let me write $x = 10^p$. Then your expression is $$ x - \frac{x^2}{1+x}$$

Note that $$\frac{x^2}{1+x} = \frac{x}{1/x + 1} = x (1 - 1/x + O(1/x^2)) = x - 1 + O(1/x)$$ so that $$ x - \frac{x^2}{1+x} = x - (x - 1 + O(1/x)) = 1 + O(1/x)$$

In your second calculation you only evaluated $x^2/(1+x)$ to within $O(1)$, not $O(1/x)$, so naturally you have an error at the end that is $O(1)$.