The main difference between the shutters of a DSLR and a smartphone is that the first is a mechanical device, whereas smartphones use a purely electronical 'shutter' (the small size of the camera in a smartphone would make a mechanical shutter hard to produce).

That said, at least some modern cameras provide a hybrid system. They have a full mechanical shutter system, and exposition starts through opening the shutter, but is ended like an electronical shutter (by timed readout of the sensor pixels). Suggested advantages are less wear on the mechanical shutter, and less noise when taking a picture (electronical shutters are silent)

At least in theory, electronic shutters can allow (much) shorter exposition times than mechanical shutters. See this article for some explanations of the different shutter systems.

The human eye has no equivalent to a shutter (the eye lids work more like a lenscap), and works in a different way from a sensor pixel. Very basically, the receptors in the eye fire when hit by a photon, and then need some time to recover. They do not accumulate photons for a fixed time.

The human eye does change sensitivity in low light conditions, which is why your eyes take a few minutes (up to 20 min) to adapt when going from a lighted room into a dark one (or outdoors at night). There is also some adaptation to the colour of the ambient light: white paper appears white under sunlight, in shadow, or under incandescent lighting, but those different light sources have very different colours (as shown when taking pictures of such a sheet with your camera's whitebalance set to a value other than 'auto')