With the proliferation of modular tools it is possible to have a tool that is both axe and hammer. You use a hammer in ice climbing to drive in pitons (A metal spike fitted at one end with an eye for securing a rope and driven into rock or ice as a support in mountain climbing – www.dictionary.com), hammer-ins and hooks. An ax-head is used to chop away surface ice to get to the solid ice.
The ax-head is called an adze. These come in different shapes and sizes and are generally used for clearing snow and chopping ice away. A flat adze is great for snow clearing and would be a good choice for the mountaineering crowd.
The hammerheads are just that heads used for hammering. They generally come in round, square or hex shapes. They all work about the same, although the hexagon can be used for rocky mixed routes.