Should I say a date differently from the way it is written?
It is
October twenty-fourth
because it's an ordinal number. It's the same as 24th October
This will vary by location, especially between the US and Commonwealth countries. In Commonwealth English the date would appear before the month as an ordinal number 24th October, and pronounced as such, because this is a short form of saying "the 24th day in the month of October". Even when I see "October 24" I tend to say "(the) twenty-fourth of October".
In the US however the convention is different; whereas in Britain days of the month are counted ("the first day, the second day," and so on) days there seem to be labelled ("day one, day two, ...") so the long form of October 24 might be "Day 24 in the month of October" or "Month of October, day 24". As such the cardinal number makes some logical sense. However, as the Grammar Girl article (and several others) point out, the convention is still to use an ordinal number ("October twenty-fourth").
However a mixture of both is appearing across the world. Certainly in the UK there are people who use the cardinal number (much to my own annoyance) and as genesis has pointed out in his answer, "October twenty-fourth" is also an acceptable pronunciation. So although the two paragraphs above give some clue as to what might be logically or traditionally correct, current form is to go with whatever you're most comfortable with as the terms are fast becoming interchangeable.