feynmf uses METAFONT which will produce bitmaps.
feynmp, on the other hand, uses METAPOST which will produce vectors.
Vectors are scalable - they scale smoothly without significant loss of quality. Bitmaps do not. This is the difference between an image in PNG or JPG format, on the one hand, and SVG or PDF format, on the other. The latter formats scale properly whereas the former do not. (Though PNG and JPG can be high-quality images, of course. But they need to be viewed at more-or-less the intended size else they will look pixellated.)
Moreover, METAFONT fonts display poorly in some PDF viwers. I'm not certain whether or not this applies to diagrams or only fonts, but if it applies to diagrams as well, you definitely do not want to use
feynmf if your documents may be viewed with such software. Since the most notable viewers falling in the 'some PDF viewers' category are Adobe applications, this pretty much means you want to avoid METAFONT if sharing your documents with anybody you cannot be sure will use an alternative viewer.
In addition, MaxNoe pointed out that
feynmp-auto enables you to automate the use of
feynmp, and giordano added that
feynmp supports colour, whereas
feynmf will complain if colour options are used.