Recognize and recognise are the very same word with two spellings, not two different meanings. Recognize is the standard American spelling and also Oxford's preferred British form, while recognise is the everyday spelling used across the rest of British English, as in "I didn't recognise her with short hair."
The core difference
There is no difference in meaning at all — only a difference in which -ize/-ise spelling convention you are following.
- recognize — identify someone or something as already known, or formally acknowledge it: Do you recognize this song?
- recognise — the same word in everyday British spelling: I almost didn't recognise her with short hair.
Which spelling should you use?
Recognize traces back through Old French to the Latin recognoscere, "to know again" — from re- plus cognoscere, "to learn or get to know." That -ize ending is the older, more historically justified form, which is why it survives as Oxford's house style even in British dictionaries, alongside American English as a whole.
Recognise follows the more common British habit of swapping -ize for -ise, seen in words like organise and realise. If you are writing for a British audience outside Oxford style guides, recognise will look the most natural; for American readers, recognize is the only safe choice.