We investigate the redshift evolution of disc- and bulge-dominated galaxies using a mass-complete sample of $\sim$14 000 galaxies from the CANDELS survey, selected with $H_{\rm mag} \le 24$, $M_{\rm stellar} \ge 10^9,{\rm M}_\odot$, and spanning $0.2 \le z \le 2.4$. Adopting an unbiased morphological classification, free from visual inspection or parametric assumptions, we explore the evolution of specific star formation rate (sSFR), stellar mass, structural properties, and galaxy fractions as a function of redshift and morphology. We find that while disc- and bulge-dominated galaxies exhibit similar sSFR distributions at $z\sim 2.4$, bulge-dominated systems develop a redshift-dependent bimodality below $z< 1.6$, unlike the unimodal behaviour of discs. This bimodality correlates with stellar mass: bulge-dominated galaxies with lower sSFR are significantly more massive and exhibit higher Sérsic indices than their star-forming counterparts, despite having similar effective radii. Based on a Gaussian mixture decomposition, we identify two evolutionary tracks for bulge-dominated galaxies: G1, a long-lived, star-forming population with disc-like properties; and G2, a quenched, massive population whose prominence increases with decreasing redshift. The evolution of the star formation main sequence and morphology–mass fractions support a scenario in which G2 systems form through merger-driven transformations of massive discs. Our results indicate that bulge-dominated galaxies are not a homogeneous population, but instead follow divergent evolutionary paths driven by distinct physical mechanisms.