The difference is how leaders are voted in and by extension, how they rule.
People's democracies and liberal democracies basically have two main different ways of doing that.

For a liberal democracy you have:

1. The two-party-system, where there's a first-past-the-post voting system so in practice only two party can be realistically be voted in.
2. The multi-party-system, where the winning party in practice always needs a coalition of parties in order to function.

The problem for these "democracies" is systemic campaign fraud that puts oligarchs in power and, in practice, for all-countries-but-one this means foreign oligarchs only and this in turn turns into a one-nation-rules-all empire, where all other national leaders are simply vassals to the oligarchs of the dominant nation.
The most blatant example of this are the concept of interim presidents, but only for non-compliant nations to the liberal democratic dominant nation of course.
I mean, do you really think you would accept an interim president of a national from your country that fled to the country choosing the interim president, let's say a US socialist that fled to Venezuela or Edward Snowden coming back from Russia?

For a people's democracy you have:

1. The vanguard democracy, where a socialist committee chooses a candidate and have just one person on the ballot, people can then vote for or against this person. If voted against, the committee chooses its next candidate.
2. capillary democracy, where you vote locally and those local leaders vote upwards until the national leader is chosen, with a socialist committee that filters out candidates through having them take civil service exams.

While it should be obvious that a capillary democracy is superior in getting people their voices met,
even a vanguard democracy solves the giant issue of systemic campaign fraud benefiting the oligarchs.