Hi Ileney,
There is common confusion over these materials. Firstly, both tooth and tusk (what you are calling ivory) can be found in fossilized and 'unfossilized' form. The fossilized items are in reality stone which has slowly displaced the original organic matter over a long period of time. I have only ever handled a couple of old snuff bottles made of mammoth tooth. They had the weight and feel of stone - heavy, cold to the touch, and so on. The same would apply to fossilized mammoth ivory - although I have never handled any personally. It is a geological process, so takes several million years for anything to become fossilized (including bones, wood, etc.).
If the mammoth tooth and ivory are sourced from animals that died only 10,000 to 40,000 years ago then items made from them will still be in their organic state. When handled you will immediately notice them to be fairly light in weight and warm to the touch. I have not (at least not knowingly) come across any organic tooth material, but mammoth ivory is seen on the market as an 'eco-friendly' substitute for elephant ivory. It looks and feels the same. Consequently it gets used for things like reading glass frames, combs and knife handles. Many eBay listings claim to be made of mammoth ivory to get around the CITES ban, but I suspect are not. Mammoth ivory is relatively expensive, while elephant ivory is not. Unfortunately it's still cheaper to slaughter elephants than dig up the permafrost in Siberia to extract frozen mammoths.
Anyway, one can easily test the difference by checking the Schreger lines:
https://en.wikipedia.org/wiki/Schreger_line Tom