I had (and lost) a large micarta as my EDC. I currently own and EDC a small annual with mammoth ivory. I love both materials, but the ivory is very special. The meeting of prehistoric and modern really appeals to me.
Other than chipping, the mammoth ivory is very resilient. It has been stabilized over tens of thousands of years and is near impermeable.
Micarta will absorb oils, and also shed them. It is otherwise very durable and has a great grippy feel. The ivory is smooth, yet still not slippery, and much more beautiful to look at.
Were I to choose a wood, it likely would be Lignum Vitae. The physical strength and resilience of the wood appeal to me. That being said some of the annual LE sebenzas with wood inlays look spectacular.
Personally I prefer:
Mammoth ivory (inner) > Mammoth ivory (bark) > Micarta > Lignum Vitae > Hardwoods > Softwoods
I prefer inner ivory for its creamy smooth appearance. I find the mammoth ivory bark a bit too busy in comparison. I'm sure many will prefer the opposite.