For Kydex:
Can be made very thin while still remaining rigid, an important consideration for a IWB holster and/or a wide/thick handgun. The less thickness and bulk added, the better. And it is much easier to reholster the gun as it does not collapse when the gun is out of it and your belt is putting pressure on it.
Impervious to moisture. It is possible, in very hot/humid weather, to sweat through a leather IWB holster.
Often lower cost than many leather holsters, although there is considerable overlap.
Will not lose its shape - hence its retention - over time.
Against Kydex:
Can cause rather rapid holster wear on finishes. However, this is generally limited to edges and corners and gets to a certain point then stops. Be sure to get it with a smooth inner surface.
Can be noisy - making a "snick" or "zing" - when drawing the gun. This could cause problems if you have to draw surreptitiously. (I don't know of any documented cases where it did, though.)
Can be uncomfortable, especially when carried IWB, being nearly as rigid as the gun itself, with no "give" to accommodate your bodily contours.
For leather:
Appearance and resulting pride of ownership. While Kydex is practical, utilitarian and functional, leather work can be beautiful.
As referred to above, it will form itself to the contours of your body over time, increasing comfort.
Against leather:
As referred to above, can be sweated through, unless it is made with a moisture barrier of some sort. (Some Milt Sparks holsters have this feature.)
To maintain rigidity for reholstering, it has to be made thicker, with a reinforced mouth adding still more thickness.
Being a natural material, it can degrade over time.
