practically the same:
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die
1 : to pass from physical life
: suffer total and irreversible loss of the bodily attributes and functions that constitute life
: EXPIRE, PERISH <
died in a fire> <may yet
die at his brother's hand> <
dying of old age> <likely
to die from lack of care>
M-W unabridged
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However, you would say
dying of old age
and not
dying from old age
Also in terms of hits at Yahoo:
353,000 for "died
of pneumonia" which seems more popular
48,400 for "died
from pneumonia"