Phasers operate by focusing and emitting a stream of fictional subatomic particles known as "rapid nadions" from an internal array of superconducting crystals called the "fushigi no umi."
Laser beams--in the real world of Earth--are a compression of certain wavelengths of pure light into a narrow beam. Based on the intensity of the light and compression, the beam can be entirely harmless, or of a deadly heat.
As a weapon, the fictitious phaser allows the wielder to stun, kill or vaporize his enemies, and by setting the phaser to overload, the weapon can be supercharged into a devastating explosion. The beam can also be split to hit multiple targets, cut through most solid materials, and even used as a welding torch.
Lasers, as they stand today, are capable of various heat-related tasks. Military-grade lasers can detonate distant explosives with lasers if given long enough, while less-intense lasers can perform delicate surgery or rectify certain skin conditions. Middle intensities of laser can weld construction materials together or cut them apart.
While functionally quite different, phasers and lasers share a number of common attributes in both the physical appearance of their discharges and in their usages.
Both phaser and the laser beams appear as lines of light in various shades of blue, red or orange, depending on their usage and intensity. Both beams also exude from their sources in a straight line to their target.
The laser is capable of light- and heat-based operations only; it can cut certain materials, weld materials together, transmit heat over great distances or be used to transmit information via light pulses.
The phaser, on the other hand, can not only perform all the same tasks as the laser, but can also stun enemies into unconsciousness and spread a single beam into different shapes or into several, equally powerful beams, feats not capable with the pure energy of a laser beam.
While gifted with the huge advantage of actually existing, in a purely hypothetical context the laser's limited functionality prevents it from besting the phaser in almost every possible arena.
The phaser is the Starfleet officer's one-stop tool for offense, defense maintenance and a host of other useful attributes, while the laser can only cut, weld or transmit information.