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Here's your bug: [bugzilla.mozilla.org...]
It's been broken for the last five years, and it's still not ging to get fixed soon. The basic problem is that Mozilla/Firefox leaves a space for the horizontal scrollbar rather than forcing the text to break. Try putting a continuous line of characters into the textarea and once the space is filled, your scrollbar will appear. See especially comment #92 in that Bugzilla entry.
I'm not sure if there is any way around this, I'm afraid...