It's because those spiders (the type that build webs on plants) can't handle active prey. As an example, I was once on a collecting trip and ran out of boxes, so I decided to put a regular worker ant in the same box as a garden spider. read more
These spiders (orb-weavers, for example) have become specialised to only eating prey that is stuck in their very sticky webs and that can't fight back. They can't attack anything. Active predator spiders can and do attack aphids if they come across them though. read more
Just a few photos from the web of spiders eating aphids. It'd be stupid to assume they don't eat them since they eat everything else they can catch. Especially smaller spiders that have just recently hatched. read more
They don't usually eat the smaller insects like aphids. The spider in the picture has caught a butterfly and is in the process of wrapping it up. There is one type of spider that can catch fish. read more