Horn/Stag/Antler has one major drawback: There are some fairly common bugs that like to eat it.
The Buffalo horn will last generations, given miminal care. Lots of 100 plus year old knives out there with stag/horn/bone handles, both fixed blades and folding knives.
Given even a tiny bit of care, leather washers can also last longer than you. There are WW1 knives with the original leather washers that are still as good as new.
Admitedly, some WW2 and later knives with leather need the leather replaced, if/when the varnish or shellac wears off and they are not cared for.
If using in cold/damp, leather will be warmer to the touch, doesn't get brittle as the mercury drops, and are not affected by high humidity as much as horn/stag/antler/bone can be.
Leather can mold/mildew. The US military currently requires a fungicide to be aplied to the leather washers of the issue knives, to help prevent mold and mildew destroying the handles. A coat of varnish or shellac (or maybe a thin coat of epoxy?) can also help prevent mold and mildew.
I suggest you do a web search for antique horn and stag handled knives, and look at the pictures (images) to see if you like how they age.
Both are a good traditional handle material, that have been used for hundreds if not thousands of years.
Both have advantages and disadvantages ... just like everything else.