Define fact. How is it different from factual belief?

Short Answer:

A fact can be defined as an actual condition in the world, a situation which is considered 
to be true and correct which has been proved and has evidence for it.

Factual belief: a belief that can be proved to be accurate. 

Long Answer:

A fact can be defined as an actual condition in the world, a situation which is considered to be true and correct which has been proved and has evidence for it. For example, "The world is rotating on its own axis” is a fact. 

The adjective factual means true, corresponding with reality, correct. One should separate the factual statements from the lies. The result of a natural calamity like Tsunami is a factual account of an event whereas the incidents in a novel might be a fictional story created by the writer unless it is mentioned as a real life story. Factual belief: a belief that can be proved to be accurate. Factual belief takes some time to be proven as true. 

According to Richard Layard fact is a statement that has been proven by scientific research.