The Mammoth's ice breath attack is identical to blasts from the Freeze Ray, causing the player and any other enemies to instantly freeze solid and become immobile.
This does no damage directly, but the block of ice will shatter if it takes damage or falls any distance, instantly killing anything caught inside, including the Spelunker.
The ice breath has long range but only travels horizontally, so it's best to attack Mammoths from above, never entering their lines of attack.
Mammoths are best killed with ranged weapons or bombs, and move quite slowly which makes chain-jumping with the Spike Shoes quite easy as well. If you don't have any other means, placing a mine in their path is also an effective way of dealing with them. It's recommended to avoid them if possible.
Cameo in Spelunky 2
As said by the Journal entry of the Freeze Ray in Spelunky 2, the Mammoth is now truly extinct. Because of this, the only appearance of the mammoth in Spelunky 2 are their skeletons, found in The Ice Caves.