"Ancient Rome" and "Medieval Europe" are two very large and ambiguous periods and places. I don't think it is possible to give a good answer without narrowing it down significantly, especially as a person's relative 'freedom' could vary dramatically based on their social status alone.

Well, both eras would be considered patriarchic by modern standards. I think that women played a bigger role in public life and society in general in ancient Rome (for me, that includes the whole of the Roman empire until late antiquity) Roman women had pretty solid rights guaranteed under Roman law. These had mostly to do with property rights (right to buy, sell, get a dowry, inherit etc.) and family law. Rich Roman matrons managed to amass huge properties, which also bought them political influence. Records from Roman Egypt of the later periods show that middle-class women were also pretty active in the courts, suing people, buying and selling, getting a divorce etc. They were definetly not just confined within their home, doing chores and raising the children while the men handled everything else. These laws and mores were passed down to Byzantium, so again we have records of women acting on their own behalf and claiming their rights (although appearing at court in person was considered unseemly). On a social level, women in Byzantium played a more active role than their Western counterparts, at least in the upper classes, and held (informal) positions of power and influence. Empress Irene and the sister Empresses Zoe and Theodora even managed to rule in their own right.