Pretty simple - Fabric.com is a wholly-owned subsidiary of Amazon.com. However, fabric.com does it's own order fulfillment.
Other vendors can sign up to use Amazon.com in various ways, for various fees. For a fee, you can use Amazon.com's checkout services, thereby broadening your customers' payment choices. For a different fee, you can advertise on Amazon and also use their checkout services. For yet a different fee, you can run a store that is largely hands-off in terms of the product, i.e. you have the product shipped to amazon, and you make up the ad copy, then you let them sell it and do the order fulfillment.
It's a pretty good business model for Amazon, you take all the risks in terms of buying the product, and they collect from you for storing it, shipping it, etc etc.
Since Fabric.com is a subsidiary of Amazon, they use a combination of the available methods, including selling through their own website, and selling through Amazon.
One thing you might notice though, is that if you buy your fabric at Amazon, with fulfillment by fabric.com, you lose the ability to purchase in anything but full yard increments. This is because Amazon's shopping cart, as sophisticated as it is, is only set up to do unitary sales.
Fabric.com requires you to buy in 1/2 yard increments, and their website is set up to sell that way.
That accounts for some of the price differential. If every sale is a minimum 1 yard, and the sale increment is 1 yard, the seller can sell for a lower price.
Fabric.com has a 1/2 yd minimum and 1/2 yd increments. The reason for that is that the margin on a 1/2 yard cut is the minimum needed to (barely) make a profit once labor, cost of purchase, cost of packaging and all other overhead is figured in.