Context : Aims : We present a detailed study of the two Sun-like stars KIC 7985370 and KIC 7765135 , aimed at determining their activity level , spot distribution , and differential rotation . Both stars were discovered by us to be young stars and were observed by the NASA Kepler mission . Methods : The fundamental stellar parameters ( v \sin i , spectral type , T _ { eff } , \log g , and [ Fe/H ] ) were derived from optical spectroscopy by the comparison with both standard-star and synthetic spectra . The spectra of the targets allowed us also to study the chromospheric activity from the emission in the core of hydrogen H \alpha and Ca ii infrared triplet ( IRT ) lines , revealed by the subtraction of inactive templates . The high-precision Kepler photometric data spanning over 229 days were then fitted with a robust spot model . Model selection and parameter estimation are performed in a Bayesian manner , using a Markov chain Monte Carlo method . Results : Both stars came out to be Sun-like ( G1.5 V spectral type ) with an age of about 100–200 Myr , based on their lithium content and kinematics . Their youth is confirmed by the high level of chromospheric activity , which is comparable to that displayed by the early G-type stars in the Pleiades cluster . The Balmer decrement and flux ratio of the Ca ii -IRT lines suggest that the formation of the core of these lines occurs mainly in optically-thick regions that are analogous to solar plages . The spot model applied to the Kepler photometry requires at least seven enduring spots in the case of KIC 7985370 and nine spots in the case of KIC 7765135 for a satisfactory fit of the data . The assumption of longevity of the star spots , whose area is allowed to evolve in time , is at the heart of our spot-modelling approach . On both stars the surface differential rotation is Sun-like , with the high-latitude spots rotating slower than the low-latitude ones . We found , for both stars , a rather high value of the equator-to-pole differential rotation ( { d } \Omega \approx 0.18 rad d ^ { -1 } ) which is in contrast with the predictions of some mean-field models of differential rotation for fast-rotating stars . Our results are instead in agreement with previous works on solar-type stars and with other models which predict a higher latitudinal shear , increasing with equatorial angular velocity , that can undergo changes along the magnetic cycle . Conclusions :