In the UK, all banks used to issue their own banknotes (hence the name), which would be exchanged for coinage at any of their branches. Acts of Parliament in 1708 and 09 removed that right from any bank of more than six partners in England and Wales, with an exemption for provincial bankers. This essentially began the monopoly of the Bank of England. The Country Bankers' Act 1826 restored issuance rights for banks of over six people, provided they didn't conduct activities within 65 miles of London. Several banks then issued their own notes, but the Bank Charter Act 1844 prohibited any new banks taking up issuance, and removed the right from any bank that merged with one without the right. The normal attrition started, and Lloyds Bank issued the last private note in England in 1921. Since then the Bank of England has had the monopoly on note issuance.
Meanwhile, in Scotland, the Bank of Scotland - which unlike the BoE was not government-sponsored - was given a monopoly over money in the country for 21 years after its founding by Act of Scottish Parliament in 1695. Scottish money was a mess at the time, and it was hoped that the new bank would get things in order. Immediately it set about issuing banknotes, as Scottish coins weren't in wide circulation, nor were they of any certain value against other currencies such as English or Flemish, which caused huge trade issues. The convenience of the new notes compared to the shambles of the old system meant Scotland was one of the first countries to permanently adopt notes by choice rather than as a temporary measure in a bad economy.
When the Bank of Scotland was joined by the not-at-all-confusingly-named Royal Bank of Scotland in 1727, the new bank, too, immediately issued a full range of notes. While England was issuing nothing below a £50 at the time (thus most English people would never see a note let alone own one), RBoS was issuing notes for 20s, £5, £10, £20, £50 and £100. Clydesdale Bank also issued its own notes from opening in 1838.
When, in 1826, an Act prohibited the circulation of any note below £5 in England and Wales, Scottish cultural and political figures mounted a campaign to save Scottish notes - with the £1 being the one most used by general populace. It became a point of cultural identity with Sir Walter Scott writing an essay on the importance to Scotland of having its own banking. The campaign worked and a commission reported that Scottish and Northern Irish note issuance should be unaltered.
In 1845 a reform was made in the wake of economic turmoil in the UK. The Banknote (Scotland) Act limited Scottish banks' free issue (meaning unbacked by coin or security) of notes to the average circulation of the year ending 1st May 1845. Today that means they can issue only £3m of notes themselves. The rest of the £3,000m they issue is backed by deposits in the Bank of England, using the £1 million-denomination "Giant" and £100m "Titan" notes that never leave the BoE vault.
So it was initially a completely different currency, then became absorbed into the GBP, but for cultural reasons has been kept on.