Pounding rain from a tropical storm swelled rivers and sent walls of water crushing into two southern Philippine cities in the thick of night, killing at least 521 people, many caught in their beds, officials said Saturday.

Doe from mean the same thing as "from" in "He died from working too much."?

Roughly, yes. The storm is the source of the rain; the overwork is the source of the death.
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"From" has many meanings and can be used in many ways. In, "Pounding rain from a tropical storm...", "from" means something like "coming out of". In, "He died from working too much.", "from" means something like "as a consequence of."
Thank you, Mister Micawber and Anonymous.