Apioceridae & Mydidae are placed in the ‘orthorrhaphous’ Brachycera within the monophyletic Asiloidea. A number of publications dealt with morphological characters that delimit both taxa.
Check out the Anatomical Atlas of Flies provided by CSIRO Australia for morphological terminology here.
Woodley (1989) was unable to find any autapomorphic features for Apioceridae, but postulated two autapomorphies for the Mydidae:
ventral armature of moderately to very stoutly thickened spine-like bristles on hind femora
maxillary palpus one-segmented
Yeates & Irwin (1996) found a number of autapomorphies for Apioceridae as well as Mydidae.
Apioceridae
two antennal flagellomeres, the first pear-shaped or elongate-oval, the second small and cylindrical
line of desclerotization at apex of labrum
postoccipital suture slightly arched between occipital foramen and post occipital suture
sternites 5–7 overlapping tergites
phallus needle-shaped
female sternite 10 well sclerotized and elongate
Mydidae
subgena simple or with wide median projection
maxillary palpus one-segmented, often reduced to a small nub
maxillary palpus without pit (reverses in Anomalomydas Papavero & Wilcox)
proprecoxal bridge present
macrosetae on scutum and scutellum reduced or absent
tergite 8 in males with posterior margin deeply concave (note: not in Syllegomydas Becker, 1906, which is considered to be a rather apomorphic mydas fly by the authors)
hypandrium fused to gonocoxite or entirely absent
gonostyli absent
lateral aedeagal apodemes absent
tergite 9 in females extending ventrally about half as far as acanthophorites
There are many more diagnostic morphological characters that are important for identification of species or genera. A detailed review is in preparation.