AT-OEMO-FED1 2022 Problem 1

Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?

(Walther Janous)