Elephantidae is a collective family of mammals under the classification of being large and having tusks.With that definition mammoths,elephants, and mastodons are surely part of the elephantidae
nipple  teeth
nipple teeth

family. Almost.
Being part of the elephantidae is about one of the only things that they have in common. mastodons are the oddballs. instead of being a grazer like mammoths and elephants, they’re browsers. They also have nipple shaped teeth while elephants do not. They are also different in their skeletons. Mastodons have larger flatter skulls than mammoths. They have more robust and stockier skulls than mammoths as well.
stocky robust skeleton
stocky robust skeleton

so in the end, mastodons were the ice age’s outlier of the elephantidae family.

By AlejandroLinaresGarcia (Own work) [GFDL (www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0-2.5-2.0-1.0 (www.creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons



By Jayu from Harrisburg, PA, U.S.A. [CC-BY-SA-2.0 (www.creativecommons.org/licenses/by-sa/2.0)], via Wikimedia Commons