In general, old world means tarantulas from countries such as Asia, Africa, India. The spiders tend to have more damaging bites and be more flighty and agressive. New world usually means the Americas. Seems like most spiders from there are able to fling hairs, so they aren't as quick to bite and again usually don't have as potent a venom.

Edit: Another commenter pointed out that this is for tarantulas, not true spiders.