Lindisfarne was the ecclesiastical seat of Northumbria, a northern English kingdom. Northumbria was where the Celtic Rite Christians from Ireland met the Roman Rite Christians sent by Pope Saint Gregory the Great. The two had significant theological differences which threatened to rip the Church in Britain asunder. The two most pressing of these were the calculation of Easter and the tonsure.
I won't pretend I understand the complexities of the calculation of Easter. You are probably aware that it is among the things dividing the Eastern Orthodox and the Roman Churches. Colman of Lindisfarne, who presented the Ionian side at Whitby, was emphatic about adhering to a calculation table that Rome had dismissed as erroneous.
Even worse were the anti-fashion police. As you probably know, monks get goofy haircuts to demonstrate their submission to God. [Swimmers get goofy haircuts, often monks' tonsures, because it is fun. Then they shave their heads to swim faster. But the monks stick with their goofy haircuts their whole lives.] The standard monastic tonsure is to shave the dome, leaving a ring of hair around the side of the head. We're not really sure what the Celtic tonsure looked like; either they shaved the front of the head (everything forward of the ears) or they shaved and arc from ear to ear. That's my personal favorite, but I wasn't really fast enough to bother shaving when I was a swimmer.
The King of Northumbria judged the Synod of Whitby and wisely threw in with the Roman rite. Colman withdrew all the Celtic clergy to the island of Iona, taking some of Aidan's holy relics with him. No need for hangings and burnings -- demography settled the rest of the question in favor of Rome.