Q:

Unbiased vs Impartial difference

What is the  difference between the words "unbiased" and "impartial" in terms of formality, usage, meaning?

A:

Comparing the definitions:

unbiased
adjective
Showing no prejudice for or against something; impartial:
his assessment of the benefits and drawbacks was unbiased 
impartial
adjective
Treating all rivals or disputants equally:
ODO

Favoring one party in a dispute over another party in a dispute, is a specific expression of bias. Like partial is more specific than biased, impartial is more specific than unbiased. A person could be impartial and still harbor other cognitive biases that influences their judgment:

Cognitive biases are tendencies to think in certain ways that can lead
  to systematic deviations from a standard of rationality or good
  judgment

As an example:
Judge Jerry suffers from an extrinsic incentive bias. When he sits on the bench to judge between litigants, he can be impartial, because his bias expresses equally for and against each party in the dispute. But his perception of motives will still be skewed by his bias in a way that influences his judgments. People who share his cognitive bias would rarely question his judgements, but those who don't share his cognitive bias will be inclined to challenge his judgements, even though they are delivered impartially.

