This is a complex question, but we'll take it step by step. First let's look at a definition of democracy. According to the Merriam-Webster Dictionary, democracy is:
government by the people, especially rule by the majority; a government in which the supreme power is invested in the people and exercised by them directly or indirectly through a system of representation usually involving periodically held free elections.
By this definition, in a true or pure democracy, everyone living in a country, or at least every citizen, would have a voice in the country's governance. In practice, this is hardly ever true.
Next let's look at the Athenian form of democracy. The word "democracy" comes from the Greek words demos, which means "people," and kratos, which means "rule." In other words, rule by the people. Athens is credited as having one of the world's first democracies.
However, its rule certainly did not include all the people, and not even a majority. Of the 100,000 citizens (that is, those with a hereditary claim to citizenship) that lived in Athens in the fourth century, only about 40,000 people were allowed to vote or participate in any other way in the governance of the city-state. The 40,000 consisted of male citizens over 18 years old. Women and children were excluded. Also excluded were about 40,000 people termed "resident foreigners" because they did not qualify for hereditary citizenship and about 150,000 slaves.
Any of the 40,000 male citizens that were qualified to participate in government could attend the assemblies, which were held about 40 times a year. Here's a big difference between modern democratic nations and Athens. Can you imagine the chaos of every eligible United States citizen attending an assembly together? Athens was able to accomplish this because of its relatively small size. Athens also had a boule, or council of 500, that met every day and handled immediate governing concerns, and another group of up to 500 men called the dikasteria that functioned as a sort of court to judge crimes.
When we turn to "modern democracies," the question becomes confusing. According to statistics compiled at the Pew Research Center, as of the end of 2017, 96 countries had some type of democratic form of government, but these governments vary widely in their practical application of democracy.
For instance, in a direct democracy, citizens vote for policies directly without representation. Switzerland is an example of this. In representative or indirect democracies, the people elect representatives to make policies. In presidential democracies, heads of state have significant leeway in enacting or preventing legislation. In parliamentary democracies, heads of state are subservient to legislatures.
It is difficult, therefore, to compare Athenian democracy with "modern democracy" because in the modern world there are so many different types of governments that profess to be democracies. However, simple guidelines to follow when comparing the democracy of Athens with any single modern democracy would include assessments of what portion of citizens are allowed to actively participate and how much the government is representative as opposed to directly handled by the people.