We present a statistical analysis of the first 300 stars observed by the Gemini Planet Imager Exoplanet Survey (GPIES). This subsample includes six detected planets and three brown dwarfs; from these detections and our contrast curves we infer the underlying distributions of substellar companions with respect to their mass, semi-major axis, and host stellar mass. We uncover a strong correlation between planet occurrence rate and host star mass, with stars M $>$ 1.5 $M\odot$ more likely to host planets with masses between 2-13 M and semi-major axes of 3-100 au at 99.92% confidence. We fit a double power-law model in planet mass (m) and semi-major axis (a) for planet populations around high-mass stars (M $>$ 1.5M$\odot$) of the form $d2Ndmda\propto m\alpha a\beta$, finding $\alpha$ = -2.4 $\pm$ 0.8 and $\beta$ = -2.0 $\pm$ 0.5, and an integrated occurrence rate of $9-4+5$% between 5-13 M and 10-100 au. A significantly lower occurrence rate is obtained for brown dwarfs around all stars, with 0.8$\mathrm{<mrow\; data-mjx-texclass="ORD">-0.5</mrow><mrow\; data-mjx-texclass="ORD">+0.8</mrow>}$% of stars hosting a brown dwarf companion between 13-80 M and 10-100 au. Brown dwarfs also appear to be distributed differently in mass and semi-major axis compared to giant planets; whereas giant planets follow a bottom-heavy mass distribution and favor smaller semi-major axes, brown dwarfs exhibit just the opposite behaviors. Comparing to studies of short-period giant planets from the RV method, our results are consistent with a peak in occurrence of giant planets between ~1-10 au. We discuss how these trends, including the preference of giant planets for high-mass host stars, point to formation of giant planets by core/pebble accretion, and formation of brown dwarfs by gravitational instability.