How do the Balkans have such distinct cultures from each other if they were ruled by the Byzantine Empire for centuries. Also, how does Turkish exist in so much of anatolia if it was ruled by the byzantines and Romans for even longer than the Balkans in some parts?
Because they were not ruled by the Byzantine Empire for at least a couple of the most recent centuries.
For example Bulgaria has been a separate state(s) from 7-11 centuries, and then from late 12-early XV-th. Serbia has been a state (or several states and principalities) in approximately the same time frames. We also have Montenegro, Albania and tons of different independent and semi-independent rulers at different point. During this entire time there has been influx of multiple different peoples and cultures, including slavs, turks, bulgars, wlachs, pechenegs, magyars, mongols and so on.
The situation of Anatolia is similar - while it has been Byzantine for quite a while,that was quite a while ago and the area is also a melting pot of multiple different peoples and ethnicities. Furthermore, for the past 8-9 centuries it has been dominated by turkic people - be they seljuks, ottoman and so on. Note also that the Ottoman empire dominated the region since the 14-th century.
So while the Roman and Byzantine empires have had control the area for some time, that was quite a long time ago, not for that long, and the ethnic, political and cultural composition of the region has changed a lot since then.