I wasn't aware the latter was even a real word, but apparently it's just a
much, much less common term for the
same thing.
I don't think its a spelling difference between AmE and BrE as much as it's just a two terms for the same thing, one of which is used far less. It's hard to find 'desalinization' without Google trying to assume you're misspelling 'desalination'. Switching the Ngram corpus between BrE and AmE doesn't change the results much. If if was merely a spelling difference, I'd expect to see the relative usage flip-flop as one switched between the different corpora. In both, 'desalinization' is barely used.
I like the third option offered
here - 'desalting'.