Realise and realize are different spellings of the same word, and both are used to varying degrees throughout the English-speaking world. Realize is the preferred spelling in American and Canadian English, and realise is preferred outside North America. The spelling distinction extends to all derivatives of the verb, including realised/realized, realising/realizing, and realisation/realization.
Although realize is now regarded by many in the U.K. and Australasia as the American spelling, it is not an Americanism. In fact, the -ize spelling variant is older than –ise—realize predates the United States and Canada by nearly two centuries—and has been the preferred spelling throughout most of the word’s history in English. If we can believe the ngram below, which graphs the use of realize and realise in British books and journals published between 1800 and 2000, realise had a brief ascendancy in British English from the late 19th century through the early 20th, but realize was preferred before around 1875 and is again preferred today—perhaps because of the influence of dictionaries like Oxford, Cambridge, and Collins, which encourage -ize over -ise.
But the British preference for realize is not in evidence when we search for the two spellings in 21st-century British news publications, where realise is about ten times as common as realize. We have trouble explaining this, but perhaps it’s simply that a few influential British news organizations have collectively adopted the newer spelling, while most of the publishers of books and journals (including prestigious British scientific periodicals such as Nature and The Lancet) in Google’s Books index have kept –ize. The favoring of -ise may also have something to do with a decline in classical education; with Latin and Greek no longer mainstays of British schooling, their influence on English is weakening (-ize has its origins in Ancient Greek).1
As the –ise spelling grows in popularity, realize is increasingly associated with North American writing, giving rise to the belief that it is an Americanism and hence to be avoided. There is no doubt, however, that both -ize and -ise are acceptable in British, Australian and New Zealand English, and writers in those varieties should not feel obliged to shun -ize, nor should they abandon -ise if they prefer it. The caveat on this advice is the -ize suffix does risk distracting non-American readers with what is (mistakenly) perceived to be a Americanism.
Whichever suffix you use, it is wise to use that one consistently throughout any given text. Note too that, for etymological reasons, some words are never spelled with a z in any variety of English, e.g. surmise, improvisation, televise, surprise, etc.
Just to satisfy curiosity, below is the corresponding ngram graphing the two spellings in American books and journals from the same period: