In the U.S., it’s not uncommon for the terms English and British to be used interchangeably to describe anything involving the realm that gave us such gems as Pride and Prejudice, Paddington Bear, and Benedict Cumberbatch.
But 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 Ireland is its own separate nation—the Republic of Ireland, formally established in 1949.
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.
Ireland is slightly more complicated. If someone hails from the Republic of Ireland, they’re simply Irish. As Northern Irish people qualify for British citizenship, however, they might consider themselves British as well as Irish. Plus, since the United Kingdom doesn’t have its own adjective, British can also be understood to mean “of the United Kingdom”—and Northern Ireland is of the United Kingdom.