The major difference is that the Greek democracy gave more power to the people, whereas Roman laws were made by their elected officials ( or Senate) and carried out by government officials like consuls and censors, less so in the hands of the people. The right to vote in both ancient Athens (Greece) and Rome was limited to adult male citizens. Women and slaves had absolutely no say in political matters, which is the biggest similarity, mainly do to the fact Rome was Republic, and Greece was leaning more towards Democracy.
I hope this helps!