In addition to Erik's answer I might be able to give you some insights on some disadvantages. I've been growing up around parachutes and until about 10 - 15 years ago you would often see people having a parafoil as a main canopy and a round one as a reserve (today it's very different, but that's not really relevant).
Round canopies are much simpler than parafoils and their openings have a higher chance of being successful. When a parafoil opens, the object which is attached to it should be stable and still. This means that the object can not have any forward movement, side movement or up and down movement. The object should also have the parachute having on top. If a parafoil is opened with any of the above factors present, chances are high you will end up with a (small) malfunction. A person under a canopy can easily "debug" these malfunctions, machines however can't. While round canopies can be susceptible to these malfunctions, they tend to be a lot more forgiving than parafoils.
Because of the shape and nature of a parafoil, twists and line-overs can happen during deployment of the canopy (I'm not taking into account malfunctions generally associated with bad packing of the canopy). With a round canopy (line overs don't really exist) this would not cause any trouble as the canopy would still be symmetrical. With a parafoil you will make one of the corners or sides unstable (this is how parafoils are steered), this will result in the canopy starting to pitch down to one corner and start turning. If the malfunction is not resolved the canopy will start spinning down and picking up vertical speed, ultimately resulting in a high speed crash.
In normal skydiving you would just cut away the canopy and deploy your reserve chute. I guess this could be implemented in spacecraft as well, but due to the weight of the spacecraft the vertical distance covered between start of the parachute deployment and deployed canopy might take a few thousand feet. Take into account that people want to reduce the weight of these things as much as possible, so another canopy this size might be around 100-200 kg in extra equipment for a reserve chute.
So in short:
- Higher chance of successful deployment
- Less amount of possible malfunctions
- Round canopies are more forgiving than parafoils