They mean the same thing — the end of a period of validity. Expiry is the usual British form (expiry date), expiration the usual American one (expiration date). Neither is more correct; match your audience. Expiration has one extra sense in American use: the act of breathing out.
The core difference
A straightforward variety split with one extra meaning on one side.
- expiry — British English: expiry date, on expiry of the lease.
- expiration — American English: expiration date. Also the medical sense of breathing out.
The other sense
Expiration is used in physiology for exhaling — inspiration and expiration being the two phases of breathing. That older sense connects to the word's root: Latin exspirare, to breathe out, and thence to the sense of a final breath, and thence to anything coming to an end.
Best-before and use-by
Neither word is the regulated term on food packaging in the UK, where use by concerns safety and best before concerns quality. That distinction matters more than the expiry/expiration one.