Since these are 3 completely different weapons, the blaster, photon gun and a phaser... they all go "pew pew" and shoot out a laser, so what is the difference between them? Is there any advantage to having one over the other?
Star Trek Phasers are basically coherent energy weapons, closely analogous to lasers.
Star Wars blasters were an evolved form of laser weapons, and fired a coherent energy particle "bolt", instead of a full-on beam.
From Wiki on Star Trek Phaser:
Originally (from the production notes to TOS), the Phaser was a PHoton mASER, since at the time of writing the Laser was a relative unknown, and powers were not expected to be very great. Masers, on the other hand, were already very powerful machines which produce very destructive radiation pulses. The term "phaser" has since been revised as a backronym for PHASed Energy Rectification, though from a physics standpoint even this is of equal semantic content—ordinary incoherent light is not "rectified", or synchronous, whereas Lasing and Masing emissions are rectified, or synchronous. Phasers release a beam of fictional subatomic particles called "rapid nadions", which are then refracted ('rectified') through superconducting crystals. Given the nature of photons the first acronym seems more accurate. The Star Trek: The Next Generation Technical Manual indicates that the superconducting crystals used in phasers are called fushigi no umi. This was an homage to the 1990 anime series Fushigi no Umi no Nadia, known in North America as Nadia: The Secret of Blue Water. The phasers that appeared in the 2009 reboot Star Trek appear similar to classic phasers, but fire 'bolts' of energy instead of sustained beams, and seem to only have two settings, stun and kill.
From Wiki on Star Wars blaster:
In the story line of the movie, common blaster weapons were said to use high-energy gas as ammunition; activated by a power cell and converted into plasma. The plasma was then released from a magnetic bottle effect to fire through collimating components as a coherent energy bolt. Inherent instabilities were said limit the ability to precisely aim a blaster bolt, but knowing one's weapon well could strengthen marksmanship. Plasma energy was said to be dissipated as the bolt traveled, limiting range as the energy became incoherent. The science fiction further described that longer range was achieved by using longer collimating tubes, to align the plasma energy carrier waves more closely through additional galvan circuitry.
A "phaser" according to most in-canon sources is basically a powerful polarized laser. In the reboot, the visual effects have changed such that phasers no longer fire beams but "packets" of photon energy.
A "photon gun", in general, is any weapon that fires a directed stream of photons.
A "blaster" is a weapon that fires not light, but a self-contained plasma charge. Blaster bolts, unlike phaser beams, travel at sublight speeds.