The pronunciation is exactly the same for both spellings, but the US convention tends to write it as you hear it. You hear 'z', so you write 'z'. And yes, this happens to many many English words.
What's important is not to mix the styles:
"I apologize, I didn't realize that." (US English, perfectly OK)
"I apologise, I didn't realise that." (UK English, perfectly OK)
"I apologize, I didn't realise that." (Big mistake!! You sound confused.)