The Congressional Research Service published a document back in 2012 that explains the difference. They describe authorization as:
An authorizing measure can establish, continue, or modify an agency, program, or activity for a
fixed or indefinite period of time. It also may set forth the duties and functions of an agency or
program, its organizational structure, and the responsibilities of agency or program officials.
They describe appropriation as:
An appropriations measure provides budget authority to an agency for specified purposes. Budget
authority allows federal agencies to incur obligations and authorizes payments to be made out of
So, basically, an authorization gives an agency the legal authority to perform a type of activity; however, it does not provide any money to spend on the activity. The US Constitution states that in order for the government to spend money on anything, Congress must first provide funds through appropriations legislation.
If you read a bit further in the report, in practice the waters get a little bit muddied. Authorization legislation will sometimes provide guidance on the amount of money to spend on an activity, but that does not obviate the need for an appropriation to actually fund it. Moreover, such guidance is not legally binding (i.e., it can be overridden by an appropriations bill), but it does appear to be enforced in the House and Senate rules of procedure:
First, the House and Senate place restrictions on
appropriations for agencies and activities not authorized by law. The House (Rule XXI, clause 2)
prohibits any appropriation, whether in a reported appropriations bill or offered as an amendment,
for an expenditure not authorized by law. The Senate (Rule XVI, paragraph 1) prohibits floor
amendments proposing appropriations for an agency or activity not authorized by law, with
certain exceptions. In contrast to the House, the Senate does not prohibit committee amendments
or measures reported by the Appropriations Committee from including an appropriation for an
agency or activity not authorized by law. Second, the House (Rule XXI, clause 2) and Senate
(Rule XVI, paragraphs 2 and 4) prohibit the inclusion of legislative language (such as an
authorization) in an appropriations measure. Third, the House (Rule XXI, clause 4), but not the
Senate, prohibits appropriations in authorizing legislation.
So, although authorizing legislation is not strictly necessary, it's a lot harder to get an appropriation through the legislative process for an activity that has not been previously authorized.