I am currently trying to learn the two related concepts of the Rao-Blackwell theorem and the Lehmann-Scheffé theorem. My question relates to example 2.16 from this document.

Assume we have the random sample $X_1, \dots, X_n$, where $X_i$ are i.i.d. $N(\mu, \sigma^2)$ with $\sigma^2 < \infty$. I have that $E[S^2] = \sigma^2$, where $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ and $\bar{X} = \sum_{i = 1}^n \dfrac{X_i}{n}$. My understanding is that, using Lehmann-Scheffé, we get that $E[S^2 \mid \bar{X}] = \bar{X}$. Based on what I've read, the two above theorems imply that, if we define, say, a sufficient statistic $T_1(\mathbf{X})$ and a complete sufficient statistic $T_2(\mathbf{X})$ for some parameter $\varphi$, then, under some circumstances, we can say that $\text{Var}(T_1(\mathbf{X})) > \text{Var}(T_2(\mathbf{X}))$. However, I'm having trouble understanding this last part. I've read over various notes on the subject, but I'm still not sure I understand what it's saying. Why can we say that $\text{Var}(T_1(\mathbf{X})) > \text{Var}(T_2(\mathbf{X}))$? And what are these 'circumstances' that make this inequality valid?