Also, "perilous" would not make sense to me in describing an object. For example, "WARNING: Dangerous current levels -- do not touch!" for an electrical device would look very strange as "WARNING: Perilous current levels..."
"Perilous" to me seems to be used more often about something that presents a danger to humans as they are doing something. "Traversing the perilous falls", "Walking along the edge of the perilous cliffs", or "an account of or perilous journey through the jungles of Africa". As b1947420 said, it's less common than "dangerous".
Dangerous is an action word, and perilous is simply descriptive. People are dangerous because of what they do-- you can't call a person perilous.
Danger is specific, vivid and concrete-- peril is more general and even vague. A "situation" is perilous, "circumstances" are perilous. "We live in perilous times" does not refer to specific dangers or imply same.
The people who simply say they are synonymous have a point, of course-- but dangerous is the more interchangeable of the two words.
Perilous is also getting a patina of sorts-- it is quaint, verging on obsolescence. "The Perils of Pauline" was a silent-era serial, and the word bears connotations evocative of Pearl White. "Siege Perilous" is a set phrase from chivalric literature and lore.
I very much agree with James and Foxfirebrand. The cliff example is a good one. It is dangerous (for a person) to walk along the edge of a cliff because cliff-edges are perilous. Cliff-edges, in themselves, are not dangerous but, as FFB says, "dangerous" has the wider scope and could - and often would - replace "perilous" there in daily speech.