Both notice and announcement are used for informing people about something that has take place or will take place. They are used differently, though.
Notice is inteded only to the persons affected by the effect of the communication, the concerned person or group or persons. Notice is linked to a presumption that those affected are deemed to have been informed.
Announcement is made to the public so that everyone interested may be informed of a fact. It is generally targeted to a larger number of people, the public in general.
