English and British don’t mean the same thing—because England and Britain aren’t the same place.
To make a centuries-long story short, England is one of three countries located on the island of Great Britain, along with Wales and Scotland. The United Kingdom’s full title is the United Kingdom of Great Britain and Northern Ireland, which makes it pretty clear which nations fall under the UK umbrella: England, Wales, Scotland, and Northern Ireland. The rest of the island of Ireland is its own nation—the Republic of Ireland.
Because England is part of Great Britain, everything English is technically also British, but not everything British is also English. You shouldn’t, for example, refer to the Loch Ness Monster as an English cryptid. It (purportedly) lives in Scotland, so you can call it Scottish or British. In other words, only things from England are English; anything from England, Scotland, or Wales is British.