The two livery companies involved were the Merchant Taylors Company and the Skinners Company.
The Merchant Taylors represented tailors and Linen Armourers. Linen Armourers made the padded tunics worn underneath a suit of armor to provide cushioning and prevent chafing from the metal armor. This group grew in importance because everyone needed clothing, and war was an ongoing concern much of the time.
The Skinners represented a luxury item: fur. Fur as part of clothing was expensive to import and therefore carefully regulated. Upper class needs included ermine and sable; lower class needs had to be satisfied with rabbit and cat.
Both livery companies received their royal charters in 1327 from Edward III, who not only loved opulence but also a year later was going to gear up for what became the Hundred Years War.
Rivalries between livery companies were not uncommon, but the rivalry between these two was particularly keen for some reason. The order of preference given to livery companies was not just "on paper": it determined who preceded whom in processions. In 1484, during the Lord mayor's river procession, the Skinners and Merchant Taylors treated the occasion as a boat race. This public display of rivalry (and public disruption) needed to be addressed. The Lord Mayor, Robert Billesdon (from the Haberdasher Company) created what is called the Billesdon Award.
The Billesdon Award requires each Company to host the Master and Wardens of the other company once each year. Also, each company should alternate precedence in processions each year. So all is settled and the two companies are reconciled? Not completely: the Merchant Taylors spell the Lord Mayor's name as Billesden, while the Skinners spell it Billesdon.
This is why in the previous post they were listed as alternating in positions 6 & 7. It is believed that the Billesdon Award is the origin of the phrase "at sixes and sevens," denoting a state of confusion or disarray.
Speaking of Lords Mayor and livery companies: one of the Lords Mayor was a wealthy Mercer who accomplished a lot during his life and term in office, but is perhaps most remembered now because of a cat. I'll tell you about him (the mayor, not the cat) next time.