What's the difference between 'bigot' and 'racist'?

Although the word bigot is used especially for someone with hatred or intolerance for people based on their race or ethnicity, people can be (and are often) also referred to as bigots due to their prejudicial animosity toward people for reasons such as nation of origin, religion, disability, gender identity, refugee status, and various combinations of all of these and more.

“Archie Bunker is still the same old bigot he was a dozen years ago and he’s not about to change,” says Carroll O’Connor who plays the dyspeptic misanthrope with unalloyed glee.
— Vernon Scott, United Press International, 11 Nov. 1982

Racist, however, refers to someone displaying, espousing, or acting on racial bigotry, specifically. We define the noun racist as someone who may be described by the adjective racist, that is, someone who holds the belief that race (i.e. any one of the groups that humans are often divided into based on physical traits regarded as common among people of shared ancestry) fundamentally determines human traits and capacities, and that racial differences produce inherent superiority of a particular race. This belief in a fundamental hierarchy goes a step beyond the personal animosity and prejudice shared by bigot.

Try as they would, the scientific racists of the past failed to discover any objective criterion upon which to classify people; to their chagrin, every criterion they tried varied more within so-called races than between them.
— Barbara J. Fields, “Slavery, Race, and Ideology in the United States of America,” Racecraft: The Soul of Inequality in American Life (with coauthor Karen E. Fields), 2012

Bigot is the older of the two words in English, first used in the mid-1600s. As it did in French, from which it was borrowed, bigot then referred to a hypocrite and especially a superstitious, religious hypocrite. Racist is a word of much more recent origin, as is racism, with no citations currently known that would suggest these words were in use prior to the early 20th century. But the fact that the words are fairly new does not prove that the concept of racism did not exist in the distant past. Things may exist for a considerable time before they are given names (T-shirt does not appear in print until the 20th century, although the article of clothing existed prior to 1900).

The use of the word race that led to racist and racism dates to the late 18th century. Advances in the field of genetics in the late 20th century determined no biological basis for races in this sense of the word, as all humans alive today share 99.99% of their genetic material. For this reason, the concept of distinct human races today has little scientific standing, and is instead understood as primarily a sociological designation, identifying a group sharing some outward physical characteristics and some commonalities of culture and history.