Sensor area doesn't determine resolution in the same way as the film era. Back then simply increasing the area of film would yield a similar increase in the size you could print, and therefore the detail you captured. In the digital world sensors can have different numbers of pixels per cm

Both 12MP compact and DSLR will resolve similar levels of detail but the DSLR sensor will be about 5-6 times bigger in each dimension. The extra size tends to provide better low light abilities as well as shallower depth of field at the same f-stop. This is because at the same f-stop a small and large sensors receive the same light per unit area, however a larger sensor receives more light in total leading to a reduction in photon noise. The difference in depth of field is due to the need to use a longer focal length for the same field of view, giving a larger apparent aperture.

Important note:

Differences in light gathering ability, DOF are not absolute and depend on the actual lenses that are available. For example it is true that for a small sensor compact and DSLR with f/2.0 lenses the DSLR will gather more light and have shallower DOF, however if you compare a 35mm DSLR to a medium format camera (with larger sensor) you might find that f/2.0 lenses aren't available for the medium format, so the smaller sensor 35mm DSLR can gather more light and attain shallower depth of field due to the availability of fast lenses.

Second note:

Not all compacts have small sensors, some now have sensors the same size as most DSLRs, such as the Fuji x100, Sigma DP1, and the Sony NEX range, and some are in between - e.g. the new Canon G1X, the micro 43rds range and the Nikon 1 series.

Final note:

Whilst a compact will resolve a similar level of detail to a large sensor camera with the same number of megapixels, you would expect the large sensor to have higher peak sharpness. This is the real advantage of larger sensors. As demonstrated above light gathering and DOF stop being an advantage after a point (as faster lenses become impractical) however resolution continues to go up as format size increases. The materials themselves ultimately limit the resolving power of a camera system and thus you will only gain from having physically bigger glass and sensors.

Here's a real world example. The GigaPxl project used a cold war era satellite spy camera with a custom lens to capture images on film that could be scanned to a billion pixels. They built a custom 215mm f/22 lens. The film size was 9"x18", to get the same field of view, depth of field and total light captured on a full frame 35mm sensor you'd need to use a 18mm f/1.8 lens. Currently the closest you could get would be an 18mm f/2.8. Whilst an f/1.8 lens could probably be made at this focal length, it would have to be sharp enough wide open for a giga-pixel sensor!