As the name indicates, anti-icing systems prevent the formation of ice (aircraft surfaces) while the de-icing systems remove the ice after it is formed.
Anti-icing systems can be active or passive. In general, most of the active anti-icing systems are thermal, which use the engine bleed air routed through the pipes in wings, engine intakes etc.
Another method of anti-icing is to have resistive circuits in wings, which can be generate heat when current is passed, preventing ice formation. This can be used also in de-icing mode to remove ice, which is more energy efficient compared to the anti-icing mode.
One of the oldest de-icing method was the pneumatic one, which uses an inflatable rubber boot to remove accumulated ice from leading edges of wings.
"Deicer (PSF)" by Pearson Scott Foresman - Archives of Pearson Scott Foresman, donated to the Wikimedia Foundation→This file has been extracted from another file: PSF D-240001.png.. Licensed under Public Domain via Commons.