The springs may not have enough support underneath to spring back at you when you sit. That's what the jute webbing that khwils mentioned is for.
Here are a couple of sites I found that show you what we're talking about:
http://www.carrscorner.com/springtying.php
This page has everything I think you'll ever need to know about tying springs.
http://www.diynetwork.com/diy/ln_uphols ... 25,00.html
This DIY lesson shows them reupholstering a chair. There's a fairly good paragraph and photograph of spring tying.
I'm usually in the habit of keeping whatever I can when I reupholster anything -- if only to save a bit of money on the job. I did one chair that's very similar to yours, and I pitched every bit of stuffing and filling -- it was real horse hair and it gave DH's and my allergies a fit.
Hope this helps,
Connie