What is the history between Little Bill and English Bob in Unforgiven?

Watch Little Bill's conversation in the jail to the writer. It shows that Bill ran in the same social circles as Bob when he was younger and that there is little real difference between them. He also knows about Munny and his history.

The old gun fighters have grown up and are trying to find ways in the world but the world has no real use for them anymore.

Bod tries to become a posh English assassin (notice his real accent comes out when he is being driven away). At heart he is, basically, an East End thug but is trying to become something else and it falls apart, he is beaten, humiliated and driven out. The only part of him that still works is his gun skill (notice the competition on the train).

Little Bill was a gun slinger and is trying to become an honest man but simply does not have the skills. Notice he is trying to build a house (very common skill for a man at that point) and he is terrible at it, he is also not a very good Sherrif. His spineless way of handling the girl having her face cut up just causes trouble for the town. His bullish way of trying to stop assassins from arriving does not work. He beats Ned to death even after getting the information about who was with him, which directly causes Munny to come back and kill several people.

Munny tried becoming a farmer but his farm was failing, to make enough money to set himself up he has to resort to his gun skills and to handle that stress he resorts to drinking (notice how the first time he drinks is just before he rides back to town to kill Little Bill).

In a deeper analysis you can say that these men know each other as well as they know themselves as once you cut beneath the surface they are the same.

If I remember correctly Little Bill had a run in with English Bob in another town while serving as a lawman. Bob is a known 'assassin', Bill's label for vigilantes, gun fighters, and bounty hunters. Bob is also well known for murdering Asian immigrants serving as railroad workers, something he himself alludes too when the character is introduced. All that aside, he does bring a firearm into town, something Bill demonstrates zero tolerance for.

Or maybe Bill just doesn't like wizards ;) , I jest.

English Bob alludes to the fact that Little Bill was a drunkard and (by implication) an outlaw. They definitely crossed paths before but it is not indicated whether it was adversarial. Little Bill refers to Bob and himself as rare examples of "dangerous men" - men who can remain cool under pressure and kill without hesitation or remorse. These two and William Money contrast starkly with everyone else who either are incapable of killing, panic during fights or are consumed with guilt for their actions.

To punctuate the last point, the kid who kills one of the cowboys in an outhouse swears off violence immediately afterward. Both Little Bill and William Money murder unarmed men and go on with their business, so does English Bob who shoots defenseless chinamen for the railroad and drunkenly murders a guy who hooked up with his crush. Whatever their past, Little Bill knows Bob is the real deal and he treats him much more seriously than the unfamiliar William Money who is beaten but not jailed or run out on a rail. Bill's biggest mistake was not realizing Money is a "dangerous man" until after the assassinations.

The character of English Bob is set to contrast with William Mummy. Mummy is a real killer, English Bob a self proclaimed killer.

The reporter hears the story of English Bob and after that, he sees him be humiliated by Little Bill, what makes him a liar, at least a great story teller.

In the last scene, he sees in Mummy what English Bob supposed to be.

But English Bob liked to be an outlaw being a fraud, and Mummy who is a real brutal killer, lives in regret.

Pretty sure there is a soliloquy delivered by the Sheriff in the film where he describes all of English Bob's "achievements" as frauds. They happened all right, but should be attributed to other people. Bob claims credit to build his "fame" as a gunfighter.