Axolotls are a species of salamander. Newts are a family of salamanders. All salamanders belong to the class Lissamphibia, order Caudata. Axolotls, eastern newts, hellbenders, palmate newts, all of them are Caudates (salamanders).
Axolotls were actually considered a sub-species of tiger salamander for many years.
Class - Lissamphibia (well, this is actually a sub-class of Amphibia, depending on who you ask)
Order - Caudata/urodela (salamanders, or 'amphibians with tails')
Family - Ambystomatidae (mole salamanders)
Genus - Ambystoma
Species - Ambystoma mexicanum (axolotl)
Newts are in the family Salamandridae, making them still salamanders. Like all squares are rectangles, but not all rectangles are squares. All newts are salamanders, but not all salamanders are newts.