The Circuit is what you hike when the W is simply not enough. The Circuit is a several days-long (can be 4 to 7, with most people taking more like 7) hike, with fewer of the comforts available on the W. Whereas on the W you can sleep at a refugio every night, or even rent a tent that is set up for you when you get there, on the Circuit you must be more independent. It's a long backpacking trip around the Paine Massif, with tougher terrain than much of the W, with steep, wind-blown ascents, and tree trunk scrambling.
The intrepid will add the W to the circuit, making it from 8 to 11 days, and will want to plan well to be able to purchase food from the occasional stores, rather than dragging all their supplies with them for more than a week.
The circuit is almost always hiked counterclockwise for the purposes of hiking down, rather than up some of the trickier ascents, and so you have the better views in front of you, rather than at your back. It is much less hiked than the W, due to the length of the hike, and due to the fact that you must camp out at least a couple of nights (and bring a stove to cook, etc). However, every year the park becomes a more popular place to visit, and if you come during high season, you certainly will not have it to yourself. There are ways to add a bit more isolation to the trip, such as walking the first part of the hike at Laguna Azul (the old trail) rather than on the current trail. There's a river crossing later (you're on the opposite side of the river for this part), and you'll pay for a river crossing at Refugio Dickson. Starting the trek at Laguna Azul will require you to either hitch or arrange private transportation, and hikers must be aware of the risks they are taking in entering the park in such an isolated location. That being said, it is more likely to see wildlife on this side of the park, since the human impact is even less.
It cannot be stated thoroughly enough that the weather in Patagonia is extremely unpredictable, and can be very harsh. You must have enough dry clothing and foul weather gear (and a solid tent) to deal with driving rain, snow (even in summer) and strong, gusty winds.