Athens had a direct representation system - all male adult citizens could vote on laws. We have a representative system where we elect people to (allegedly) represent our views and vote on laws for us. The USA is also a lot larger than Athens..

Participation was limited to adult male citizens, perhaps 15% of the adult population resident in Athens. We surpassed that sometime between 1869 when we outlawed slavery and 1919 when all women got the right to vote. Britain passed the Greeks in 1832 with the Great Reform Bill.

The practice of ostracism - where citizens could vote to expel a particular citizen is unknown in our system.

Many positions of authority were chosen by lot rather than elections. Doesn't seem like a good idea, you'll say, "what if someone unqualified is given power?" And how would that be different than now?

There were many committees with duties to report to the Assembly.

There were no formal political parties, but there were many factions and shifting coalitions.

There was no separation of powers, the Assembly was supreme. There were other bodies that did have some authority. Perhaps it would be better to say that the separation of powers was different than we have.

There was no written constitution, but there was a collection of laws. Thus there was no Bill of Rights. In the Assembly, citizens had freedom of speech. There was certainly no freedom of religion, except that their religion was very polymorphous.

Legal issues were dealt with via trials but without police, lawyers and formal case law. Individual citizens could bring prosecutions on behalf of the entire citizenry.

Juries were massive, consisting of 201, 501 or even 1001 jurors. There was no formal judge with presiding powers, but their may have been a timekeeper. Jurors were paid.

There was some legal review of laws.

The entry to the Pnyx site.