First section on hero weapons, second on legendaries, third on mutual.
Hero weapons evolve based on specific requirements met and a randomized choice of traits you qualify for upon the opening of each chest. The first chest adds runes with different color based on morality, the second a design for the hilt or handle, a third the design for the barrel, head of the hammer, or blade, the fourth affects the color, and the fifth adds an aura. A complete list of traits and requirements is available at http://fable.wikia.com/wiki/Hero_Weapons and I would suggest looking at it if you wish to work towards specific outcomes, as well as saving your game first and reloading if you don't like your morph. Many of these are cosmetic only, with exceptions to firearm barrels. The clockwork barrels come with scopes which people are unsure if they improve actual accuracy and range of the firearm. Crystal, elegant, and onyx barrels give your gun a shotgun spray if you manage to get them.
Legendary weapons at the beginning of game will have an identical appearance to the initial hero weapons. Their 5 morphs are not unlocked as you unlock their three traits, but unlocked each time you improve melee or ranged weapon performance, the same as hero weapons. Unlocking bonuses by the three missions has no effect on this. While their morphs are preset, they will only appear as that section of your hero weapon morphs. I am unsure if the same trait applying to shotgun spray firearm barrels applies with legendary weapons, but believe it does.
There's rumor that multiple quick shots shorten firearm barrels, while charged shots lengthen them. These improve reload speed or accuracy respectively. Each chest opened for ranged weapons increases the amount of shots a firearm will be able to fire before reload. It is by one for each chest for hero weapons, but it has varied in my experience for legendaries. These additional unlisted and unspecified benefits, assuming the rumors are true though it's debated, apply to all firearms. Similarly, multiple quick melee strikes supposedly thin and shorten your melee weapons to move more quickly while blocking and flourishing increases it in size to strike more damagingly, according to rumor, across all melee weapons.