We know that Socrates and Plato are two of the most influential Greek philosophers in current Western society. But what influence did they have in the Greece of their time and subsequent centuries? Were some of the pre-Socratic philosophers more, equal or less influential, say, from years -300 to +300?
A good question and I'm not sure the answer is so obvious, though there is probably good scholarly research on the topic.
Socrates was written about by Plato, Xenophon, and Aristophanes and, of course, tried and condemned on the basis of his supposedly impious "influence" on the noble youth of Athens. So there was clearly local and contemporary influence. But he did not write or travel, so he was probably not as well known at the time as figures like Parmenides or Protagoras or, say, Anaxagoras, who was an influential companion of Pericles.
Plato, of course, wrote many works and travelled and founded a school. Though we can't count printed citations from antiquity, it is fair to say that the survival of copies of so many of his works (unlike those of the great dramatists, say)testifies to a large and widespread contemporary influence. But "influence" is hard to measure and it is hard to escape the long shadow of Plato's enormous retrospective influence and later adoption within Christianity.
But it is interesting to consider, I don't recall reading anything specific about Plato's contemporary fame or impact on affairs of state or reputation vis a vis contemporary philosophers, so many of whom are in fact preserved in his writings. But I would guess his founding of the Academy alone would rank him among the most influential men of his age and a prolific source of ideas and learning for centuries after.
"What influence did Socrates and Plato have on ancient Greece?" Not very remarkable: Socrates was executed and Plato's later work is, as Charles Kahn puts it, "post-socratic" in a strong sense, Socrates' role being mostly decorative.(see Kahn, Plato and the post-socratic dialogues, Cambr.2013). And very early the Academy started casting Plato as a pythagorean (a market for archaic looking fakes being thus created). Aristotle's work was eclipsed for more than 200 years but then it made a spectacular comeback and from the viewpoint of late antiquity he is probably the most important figure. Platonists did the impossible to show that he just followed Plato and even today there are people who believe their propaganda.(see Karamanolis, Plato and Aristotle in agreement?, Oxf.2006) During the hellenistic era Greek Stoicism has been the widely known synthesis with Chrysippus being a most prolific author and Posidonius probably the most respected one. However of their texts nothing survived; the romans took interest just in stoic morality while later christians fiercely attacked them.
Such historical details come to show that importance is mostly a perspective, depending on available traces; Plato's works were preserved in Byzance first of all as fine literature while some 3/4 the texts of Aristotle are lost and the remains are not really finished (or polished). Scholastics revived Aristotle more a millenium after his time and reworked as Neothomism it is still the official church doctrine. Renaissance humanists and, sometime later, German philologists upheld Plato who enjoys popularity by being much more "user friendly". Even if it is difficult to judge how biased is Diogen Laertius he is the sole source presenting a more or less full view of philosophical scene from the first centuries of CE. Plato and/or Socrates do not seem to have been particularily important.
In the case of Socrates, he certainly had enough of an influence within Athenian society, whereby the city's Political and Judicial Establishment had him arrested, tried, convicted and executed for "corrupting the youth" and apostasy. He was a very controversial figure in Athenian society and seems to have had only a single close friend.....Plato.
As for Plato, he too, was not necessarily looked upon with great reverence or respect from the average Athenian during his lifetime. His unpopularity with his fellow Athenians was probably due to his philosophical views. But, despite his lack of popularity with his fellow Athenians, Plato did establish The Academy, which stood for centuries and would become an important cultural and intellectual institution for the city of Athens-(as well as for the larger Greek world) during the later years of Antiquity.
It has been said that Plato is the most influential of all Western philosophers. No one agrees with him, but everyone starts their philosophical journey by disagreeing with him. Plato was essentially a mystic. He frames his ideas in practical language, but that's just for rhetorical purposes --the practical concepts are just to provide a way of apprehending the metaphysical ones.
Plato's most important influence, of course, was on Aristotle --even though that influence was expressed in disagreement. Aristotle's philosophy can be thought of as an attempt to take Plato's topics and find actual practical answers to them, rather than using them as teaching analogies. The result was the vehicle through which the mundane processes of Athenian life were turned into abstract cultural principles, which in turn put the Greco into Greco-Roman culture, and turned the Roman Empire into a copy of Athens, writ large.
The mystical strain of Plato's thought was transmuted into Neo-Platonism, which was absorbed into Judeo-Christian and Islamic theology, and which lives on as the Perennial Philosophy. Meanwhile, the urge to fix Plato has been the starting point for countless other philosophies from utilitarianism to existentialism.