What’s the difference between "tire" and "tyre"?

It is a question of regional usage but according to The Grammarist tire is used mainly in the US and Canada:

• Tire and tyre both mean a covering for a wheel, usually made of rubber. Tire is the preferred spelling in the U.S. and Canada. Tyre is preferred in most varieties of English outside North America.

• Of course, all English speakers use tire in the sense to grow weary.

This preference appears to be confirmed also by Ngram:

Ngram a tire in BrE vs AmE.

Ngram a tyre in BrE vs AmE.

Examples

(U.K.)

Advanced systems to monitor fuel and tyre usage give the company a clear picture of how its fleet is performing. [Guardian]

The machinery would enable the factory to produce the latest generation of car tyres, the company said. [BBC News]

(U.S.)

He said that he had just returned to Haiti from studying business in Canada, and that he was helping his father run a tire retread factory. [NY Times]

Andretti also worries about drivers who neglect the simple things, such as checking their tire pressures. [Globe and Mail]

Your web research is correct. Your friends may be correct in their social circle, but across the whole of England they are not correct. As a quick check, I looked at UK websites for vehicle servicing:

kwik-fit:

We stock a wide range of tyres

Halfords Autocentres:

Book tyres online with our quick and easy tyre size finder.