Nikon D610 has a 24.0MP Full frame (35.9 x 24 mm ) sized CMOS sensor and features Expeed 3 processor. On the other hand, Nikon D3 has a 12.0MP Full frame (36 x 23.9 mm ) sized CMOS sensor and features Expeed processor.
Nikon D610's sensor provides 12MP more than Nikon D3'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.
Below you can see the D610 and D3 sensor size comparison.
Nikon D610 and Nikon D3 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.
DxOMark Sensor Scores
Both Nikon D610 and Nikon D3 sensors have been tested by DxoMark. DxoMark scores camera sensors for color depth (DXO Portrait), dynamic range (DXO Landscape) and low-light sensitivity (DXO Sports), and also gives them an overall score. Of the two cameras that we are comparing, D610 has scored 94, 13 points higher than D3.