A proclamation is typically a very formal, official announcement by the government or by a ruler.
For example: "President Trump signed a disaster proclamation for Texas."
Proclamations are also somethings associated with kings and queens (royal proclamations).
For example: "The king proclaimed that taxes would increase."
An announcement is any kind of general public statement to a group of people.
For example:
"The host announced that the party was over."
"The teacher announced that today class would be outside."