I would argue that the most important similarity between these two systems was that they had an element of democracy. It is for this reason (among others) that we in the Western world today trace our civilization and government back to the Greeks and the Romans.
The Athenians were, of course, famous for having a democracy. They are widely seen as the first democracy in the world. However, the Roman empire was partly democratic as well. Even after the Republic fell and Rome became an empire, the Senate was very politically important. The fact that the Senate retained power gave Rome some measure of public input into politics. Another aspect of both systems that can be called democratic is the fact that laws in both systems were written. This put some limitations on what the government could do because it ensured that the laws were known to all.
So, both Rome and Athens had governments with aspects of democracy. This is the most important similarity between the two.