We're mixing apples and pears here . . . M-o-P is one material, ivory is another, completely different material. The two are in no way related. M-o-P is "made" differently, works differently, and behaves differently.
Mammoth is noticeably more porous than modern ivory. Scrimshanders tend to work mammoth a lot dryer than ivory because it sucks in the inks. Applying ink often results in a disturbing ripple appearing on the worked surface, which goes away after drying. Both react to changing climatic conditions, they are "live" materials, but they are unpredictable. I have one 20-year-old drop point hunter which has been through deserts, forests and snow and is still unmarked. I have another folder who's handle cracked three weeks after delivery.
One of the problems with modern ivory is that it cannot be transported across international borders (and some US state borders I believe). White mammoth, such as that shown on the knife earlier in this thread, looks like modern ivory. Trying to explain to a customs official the finer differences between mammoth and ivory in an airport, with a plane on the runway, could be an interesting experience. This isn't an issue with the more coloured mammoths.
By the way, there are 9 accepted types of animal ivory . . . African elephant, Indian elephant (slightly pinker), mammoth, walrus tusk, fossilised walrus tusk, warthog tusk, hippo tooth, whale tooth, and narwhal.