Generally speaking, and there are way too many variations to cover them all:
A fly cutter can cover a large and adjustable area per pass. It can leave a very nice finish, though that requires some practice, trial and error, and knowledge. Fly cutters are not usually best at removing large amounts of material per pass, but can if solidly built and used correctly. Fly cutters do not generally require lots of power to do their job. The cutting tools can be high speed or carbide, and can be shop made and ground for many fly cutters. They can be bought cheaply or can be shop made, and are pretty simple tools.
A face mill has a set diameter. It uses multiple cutters, so that requires multiple times more power to drive them. Modern face mills all use carbide insert cutters, as far as I know. They can remove lots of material quickly if the power is there. Achieving a good finish can be an issue with these cutters, because the inserts do not necessarily follow one another accurately around the face mill. That can be from design, type of cutter, accuracy of manufacture, accuracy of inserts, prior damage, etc. Some face mills have adjustable cutters, which can take some time to get set up all the same, and rigidity may be lessened, but can improve the surface finish. A lot of work goes into making an accurate and solid face mill, so they can be expensive. Face mills also often need some speeds and feeds tweaking to get a good finish. There are many styles of face mills, in different sizes, configurations, and for various intended purposes. Some are small, with only a couple or a few inserts and can be run on bench top mills. Some are 8" or more across with lots of inserts. The larger ones are expensive to buy, expensive to change out a set of cutters, and require a rigid and powerful machine to drive them. On a larger face mill, some of the inserts can be removed, which reduces the power required, but also usually reduces the surface finish quality, at least at the same speeds and feeds.