I have gotten the game since the first day, and i am currently on my 5th full character. You are correct on the cosmetic effects for the weapons, but i believe it goes further into that. Obviously the power does go up, and the end result is that ur weapons are weaker than legendary weapons, but i have been seeing that it changes the style of how your fight. I have used either a sword/rifle or sword/pistol combo on my characters, but they have all been somewhat different. The thing i noticed the most is after the 2nd or 3rd chest opened. The slow-motion actions that Ive seen have changed due to the hilt/blade morph. I have had a few of these actions be used on multiple morphs possibly, but i think it dose affect your combat actions. for example, when i got the scoped morph on my rifle, it sometimes does a "crosshair" shot, or the camera starts in the scope, and then follows the bullet into the enemy. also another one was when i got the spiked scimitar blade, my hero would throw his blade into an enemy, then run over and pull it out of the victim. I believe that these movements do have a link with the hilts and/or blade morphs. I have never been able to use this motions before I got my 2nd morph. Also I am not talking about the flourishes, Im talking about the one hit kill actions. Please let me know what you think, because, like all of us, we are all trying to figure this thing out.
edit: i do believe there are basic attacks of these, like the point-and-look attack with the hobbes or the neck snap with a human enemy. these are just a few ideas, so lemme know what you think