Delta systems are typically known for their use in pick-and-place machines. They are precise and fast. Several people have adapted this to 3D printing. You can see that, even though they are delta robots, they can use very different construction and driving methods. Some, like the RepRap Rostock (shown here) are belt driven and have a huge vertical build volume.
The RepRap Simpson (shown below) is a completely different design but appears to work as fast and precise as possible.
I'm not really sure that there is a clear advantage. Someone from one of those companies told me that it gives a bigger build volume or something but I don't get it.
I don't really see any disadvantages though.
One of the things that is not really though about in components like this is tolerances, speed of movement, and stiffness. In 3D printing you do not see much of a need, yet for high tolerance, though this will come. Once you start thinking about tolerances, there is a need to recognize that at small movements a device that has linear X and Y travel will have a discrete amount steps in its travel, and hence just like if you were to product a small circle on a low resolution monitor the worse the edges would be at the 45° points of the circle. The advantage, though of a linear travel device is that it can be stiff in all three axis.
A polar device, because it travels in a circle can produce smother results, but the further you get away from the center of rotation will cause higher errors in linear position tolerances.
The delta design, I am not too familiar with. I am not sure why it is better other than it might offer very light weight. This would allow for faster movement. For something like a pick and place machine where stiffness is not a requirement, then this would be a good thing. It may, by virtue of its setup may increase accuracy in the curvature almost being a combination of a polar and linear device. It would also have the ability to tilt the head if the Z axis of each arm is controlled independently. I will have to go and do some studying of these devices.
Once again most of these accuracy issues will not be seen as the methodology is laying down a melted plastic wire that is wider than the accuracy of the movement itself. This will mask a lot of things. It may then come down to which can build the biggest part in the smallest space.
No worries, sadly, many makers of equipment do not think of it that way either. I have seen a too many designs for CNC routers that have 2 out of 3 axis really stiff and the last one an order of magnitude less stiff than the other two. Two things result from something like that. The first is that if the more flexible axis is stiff enough for the parts that are being built, it means that the other two are overbuilt. This usually increases cost as well as weight. If your device is intended to be a table top device, then these are areas that someone could use to undercut you on price and performance.
The other scenario is if the flexible axis is not stiff enough. Here you will find that you will not be able to maintain pressure on a cutter when the forces are in that direction. This will cause parts to be in shape/tolerance in one direction, but not in another.
I have a post that I am going to try and get finished this weekend about how to get stiffness in the direction you want or need it. I see this as a general issue that really shows up quite often in many other places. Quad copters are another area that tend to suffer from poor mechanical design.