I guess I should work out how to harvest silk so it's a moot point. Giant spiders are in the caverns right?
There's spiders, and there's Giant Cave Spiders. The regular spiders are vermin, meaning they don't appear on the unit list, and only occasionally and briefly appear as a dot-or-whatever on the screen. They leave webs behind randomly (as far as I know). Mostly you'll know if there are regular spiders around if you see webs in one or more of the cavern layers. You can send your weavers to collect those webs (bringing them back to a loom). Actually, the default setting is for weavers to auto-collect all webs; I recommend turning that off, because civilian dwarves traipsing through the caverns without your knowlege tend to have a lot of Fun.
Regular spiders can also sometimes appear in areas outside the caverns. Since they're vermin, cats will also destroy them, leaving spider remains. This has a negative impact on web production.
Giant Cave Spiders are full-sized units, classified as "wild animals". They appear on the unit list when they're on the map, and disappear if they leave the map. They shoot webs as an attack, rather than just dropping them randomly.
GCS are aggressive, and will generally attack dwarves. In addition to their entangling web attack, they can inject venom, and they can bite off limbs and so forth. In a confrontation between an unarmed dwarf and a GCS, the GCS is going to win, period. In a confrontation between a hunter and a GCS, the GCS will usually win....