Q:

Do party members gain bonuses from using their "story" weapon?

In the DLC, the alien Javik introduces a particle rifle. 
Is there any reason to arm him with that instead of using the best rifle in the game (found midway through the game)? It's a pretty lame rifle all things considered.
Does it even make any difference on squad effectiveness?

A:

No, there are no extra bonuses for matching a particular character with a particular weapon.
One good benefit of equipping squad mates with this rifle is that it fires a continuous beam so its easy for you to see where the enemies are when your squad mates shoot them.

