From 1204AD: A Spanish traveller named Clavijo remarked that the Greeks of Trebizond

"arm themselves with bows and swords, like the Turks, and ride the very same way". According to Mark C. Bartusis this last phrase means "that they rode nomad-style with short stirrups, almost in a crouch, wich allowed the greater mobility in the saddle rquired of horse archery."

During the beginning of the twelfth century horse archery in the Byzantine army was in the hands of foreign mercenaries, mostly Turks. The other nomadic peoples the Byzantines encountered (Alans, Tourkopouloi, Cumans) were used as light cavalry. The only native Byzantines using the bow were light infantry troops, of which we know very little.

Source: The Late Byzantine Army: 1204-1453. By Mark C. Bartusis.

During the late Byzantine Empire the Byzantine Armies employed mostly mounted archers who were mercenaries from the nomadic peoples on the borders.

Another source (Byzantine Armies 866-1118, Ian Heath) notes how archery in the Byzantine army was in 'a steady decline' since the 8th century.

“Tacita Leo VI had cause to complain that archery has wholly been neglected and has fallen into disuse amongst the Romans. He then proposed the reintroduction of enforced archery practice. Leo also reformed the cavalry in such a wat that two men out of every five were archers and no longer carried lances." (note: Leo IV died 11 may 912)

The Armies of the Roman Empire consisted mostly of heavy infantry using offensive tactics to drive of the enemy. Archers were mostly used as support. From ‘The Roman Army at War 100BC-AD 200. By Adrian Keith Goldsworthy’:

During a battle against the Alans, a Sarmatian tribe mostly using shock cavalry:

“His [Arrian's] formation was therefore designed to counter this [shock cavalry] threat, and may not jave been used against cavalry who fought from a distance. Arrian’ legions were formed with the first four ranks armed with Pila. The first rank placed the butts of their weapons on the ground and held them firmly at an angle of 45 degrees towards the enemy. The three ranks behind the first were to hurl their pila, then the eight rank would have to throw them 15 m. merely to clear their own formation. (…) behind the the legionaries was a ninth rank of foot-archers, with both horse-archers and artillery firing over the heads of these nine ranks. (…) the auxiliary infantry were similarly supported by archers and artillery firing over them. ”

“When Mark Antony campaigned against the Parthians he included more of such troops, with the result that his infantry suffered far less. Arrians’s army in cappadocia contained an exceptionally high proportion of auxiliary archers, to protect the infanry from the heavy cavalry “showing how the Roman Army could adapt to such threaths.”

While this does not answer your question, i hope it helps with the difference in role during the specified time.