Those Slavs converting to Byzantine Christianity adopted the Cyrillic alphabet while those converting to the Latin version adopted the Roman alphabet. The alphabet followed the religion/rite. An exception is the use of the Glagolithic alphabet in Croatia to write Old Church Slavonic used in the Roman rite there.

The Cyrillic alphabet was apparently first developed in the Bulgarian empire to write the Old Church Slavonic used in their version of the Byzantine rite. This was exported to other Slavic speakers adopting the same rite.

To build on /u/roelm2's answer, I answered a tangentially-related question here, specifically the part about converting to Catholic rather than Orthodox Christianity as a power play against encroaching Byzantine and Ottonian influence in Eastern Europe.