Scottish and British may well originally be two different tribes. Although I think you're forgetting that "British" is a combination of numerous ethic groups, created over millenia. Angles, Saxons, Normans, Celts, etc.
But do a little research on the Act of Union 1707, and you will find that Scotland has been a willing constituent of the United Kingdom for quite some time, which makes them British.
Scotland, England, Northern Ireland and Wales are all part of the United Kingdom of Great Britain and Norther Ireland. Britain or UK for short. Each one is British.
On a geographical note, Great Britain is actually the name of the large island on which you will find England, Scotland and Wales - The British mainland. So Scotland is indeed British, both politically and geographically, no matter what the tribes were originally called.