A good question but almost impossible to answer briefly. I think you will have to do the reading and come to your own conclusions. The main problem is that, although we know a lot about both cities in different periods, we simply don't know enough detail to say how they well or otherwise they functioned. What was public health like, for example? What was life like for women? How much money did the ordinary citizen have in her or his pocket (so to speak)? How were children treated or regarded? As for questions of democracy and republic, these terms are not mutually exclusive but overlap. Both states developed a considerable degree of democracy, at least for men, but it can be argued that, for all the apparent political freedom, power still resided in the hands of the rich and/or aristocratic. A good example is fifth century Athens, often regarded as the pinnacle of democracy in the ancient world, and yet Pericles was able to govern almost as a king. Similarly in Rome, real power for many centuries was in the hands of a group of perhaps 15 or 20 families or clans. These questions are still asked about democracy today and you may well wonder, as I did when studying ancient history, whether human society really changes all that much in the long run.
Yes I agree with anzio45, you will have to read about both civilizations and determine for yourself which one was more effective. As you should know history is very interpretative, and different scholars would have different answers to your question based on their perceptions. Based on history itself and looking at strength, power, organization, economy, and societial advances I would say ancient Rome. The Roman empire was one of the most powerful empires in the ancient world, but also there has never been such a powerful empire since then besides Great Britain and the United States today, but these two countries cannot compare. Look at these two sources and then determine for yourself. It also depends on what effective means to you: power and might or fairness and justice?