Examples of smooth manifolds admitting inbetween one and a continuum of complex structures

Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.

For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.

In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.

Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.

There are countably many complex structures on $S^2 \times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_{2k}$ are the only options. This is the main result of

Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.

The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.