Nikon D3300 has a 24.0MP APS-C (23.5 x 15.6 mm ) sized CMOS sensor and features Expeed 4 processor. On the other hand, Nikon D300 has a 12.0MP APS-C (23.6 x 15.8 mm ) sized CMOS sensor and features Expeed processor. Nikon D3300's sensor provides 12MP more than Nikon D300's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
Another difference between these two cameras is that Nikon D3300's sensor lacks anti-alias (Low-Pass) filter. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moire occurring in certain scenes.
Below you can see the D3300 and D300 sensor size comparison.
Nikon D3300 and Nikon D300 have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with same focal length and aperture.