British spelling: "apologise." (Also, "criticise" and "realise," for instance.)
American spelling: "apologize." (Also, "criticize" and "realize," for instance.)
There is no mystery to why some people spell it one way but others the other way. Check where they're from.
British spelling: "Palaeontology."
American spelling: "Paleontology."
Again, no mystery.
I am just waiting for the same split to occur with "archaeology." The reason for the split is that the "ae" in "palaeontology" and in "archaeology" is really one of those combination-letters--the "a" and the "e" used to be written all as one letter--which can be rendered either by separating the "a" from the "e" or by simply dropping the "a" altogether. Increasingly, the "a" is being dropped from words like that. (See "aesthetic" and "esthetic.")
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