A commission is a group appointed by the government to accomplish a certain task. The commission can cease to exist when the task is completed (e.g., the 9/11 Commission, which investigated the events surrounding 9/11). Or the commission can be ongoing indefinitely (e.g., the Securities and Exchange Commission, which oversees Wall Street).
In government, committee is a group of people who meet to make a certain decision or to gather information and produce documents. Committees, too, can be short-lived or long-term.
A committee can also be a commission, and vice-versa. Because the term "commission" is more specific than "committee," a group of people that fit both descriptions would usually call themselves a commission. For example, when a crisis affects a city or state, the mayor or governor will usually appoint a commission to gather information, make decisions/recommendations, and accomplish certain other tasks.