The canonical partition function $$Z(\beta)=\sum_i e^{-\beta E_i}$$ normalizes Boltzmann probabilities, but it also packages the thermodynamic consequences of the entire energy spectrum.

The probability of state $i$ is $$P_i=\frac{e^{-\beta E_i}}{Z},$$ so the mean energy is $$\langle E\rangle=\sum_i P_iE_i. $$ Differentiating $Z$ gives the compact identity $$\langle E\rangle=-\frac{\partial\ln Z}{\partial\beta}.$$

For a two-level system with energies $0$ and $\Delta$, $$Z=1+e^{-\beta\Delta},$$ and therefore $$\langle E\rangle =-\frac{\partial}{\partial\beta}\ln(1+e^{-\beta\Delta}) =\frac{\Delta e^{-\beta\Delta}}{1+e^{-\beta\Delta}}.$$ At low temperature the mean energy approaches zero; at high temperature it approaches $\Delta/2$.

The Helmholtz free energy follows from $$F=-k_BT\ln Z.$$ Once $F(T,V,N)$ is known, entropy, pressure and other equilibrium quantities follow by differentiation. The heat capacity can similarly be obtained from the temperature dependence of $\langle E\rangle$.

The partition function is therefore more than a normalization constant. It is a generating object that translates microscopic energy levels and degeneracies into macroscopic thermodynamics. Different physical models enter through their spectra; the statistical machinery remains the same.