It's very important to remember that, prior to Marathon, the Persians had largely been victorious against the Greeks and that, even at Marathon, the Greeks didn't exactly have an easy time of it against the Persians. As /u/gynnis-scholasticus has said, the Persians broke the Athenian centre at Marathon and came very, very close to winning the battle.

Equally important is the fact that the Greeks and Persians at Marathon and Plataea most likely had roughly similar numbers, rather than the Persians having significant military superiority. This isn't a new discussion - Hans Delbruck made this argument as far back as the 1880s, although it's been argued with much better evidence since, and that view is fairly well accepted now in the recent historiography. There are a number of reasons for this, from estimates of the available water only being sufficient to support ~16 000 Persians to the fact that there's good evidence to infer that the Athenians would have had as many light infantry as hoplites (18-20 000 total) in the battle. At Plataea, the dimensions for the Persian army camp (as given by Herodotus) similarly suggest an army not much bigger - if at all - than the Greek army.

I won't go too much into the relative merits of Greek vs Persian infantry - the posts by /u/iphikrates that have already been provided go into this - but I will briefly discuss the significant differences between Marathon and Agincourt, because it's the differences in terrain and circumstance that resulted in an English victory at Agincourt and a Persian defeat at Marathon.

To begin with, Agincourt was a narrow field - with perhaps 6-700 meters of width of useable terrain depending on where exactly the fighting happened - that prevented either army from properly deploying. If we accept the traditional figure of 5900 for the English (which I do, although Anne Curry has argued for closer 9000 men), then the English would need almost 1500 meters of space to fully deploy^1 . The French, deployed more deeply and with a more complex battle plan, needed even more space (especially for the cavalry).

The narrowness of the terrain effected the French far more than the English. Instead of deploying in a single line, with the men-at-arms in the centre, archers and crossbowmen on each wing and cavalry beyond them, the French were forced to deploy their men-at-arms and gros valets in three battles^2 , with the archers and crossbowmen between the first and second battles. As a result, each French battle was numerically smaller than the total English force, and the need to keep space on either side of the vanguard for the cavalry to operate in meant that the vanguard likely attacked on a narrow front.

In addition, the rain had been very heavy in the lead up to the battle and the grooms had been exercising the horses in front of the French camp the night before the battle. The soil at the battlefield is particularly clingy as well. The net result was that well armoured cavalry intended to break the English archers were unable to move faster than a walk or very slow trot and couldn't close the distance before the psychological impact of the arrows broke their morale and caused them to retreat. This retreat, in part because of the panicked horses and in part because of the terrain, funnelled some of the cavalry through the French vanguard, breaking it up further (the mud had already impacted cohesion) and rendering them vulnerable to the English archers swarming out from their lines and overwhelming the exhausted, battered and panicking French men-at-arms^3 . The main battle of the French arrived after the vanguard had more or less been defeated and, because of the remnants of the vanguard and the mud, soon suffered a similar fate.

The battle might have been very different, however. The French had sent a small force of cavalry around to attack the English from behind, and it did manage to get in among the baggage before the wagons had properly formed a defensive perimeter behind the English lines. If not for the mud, the French armoured cavalry would almost certainly have broken through the archers on each wing, as they did at Valmont in 1416 and Verneuil in 1424, and with the baggage already under attack or taken, they may have then reordered themselves to attack once again, ensuring that the archers could not engage the advancing vanguard (or at least not the same extent), while the vanguard itself would have been able to advance more quickly and in better order.

In so far as any analogies can be made for two battles fought 1900 years apart and with vastly different social and military systems, the Greeks at Marathon could be said to have had a similar plan to the French before and at the start of Agincourt. Just as the French had hoped to nullify the English archers by charging heavy cavalry through them, the Greek charge at Marathon greatly reduced the effectiveness of the Persian arrows and rapidly forced the situation into hand to hand combat. This rapid charge may also, as Peter Krentz has suggested, have begun the battle before the Persian cavalry could deploy and prevented them from playing a role in the battle.

Unlike the French at Agincourt, the Greeks were a relatively coherent force and, although they lost in the centre (a combination of it being thinner - although whether deliberately or a natural result of the army spreading out to prevent being outflanked will never be known - and the better Persian infantry being the centre), their wings retained enough presence of mind not to pursue their enemies very far but to recollect themselves and set out after the Persian centre as it pursued the broken Athenian centre.

Further Reading

My main sources for Marathon itself were Peter Krentz's The Battle of Marathon and Roel Konijnendijk's 'Neither the Less Valorous Nor the Weaker': Persian Military Might and the Battle of Plataia. For the Greeks and Persians more generally, George Cawkwell's The Greek Wars: The Failure of Persia, Sean Manning's Armed Force in the Teispid-Achaemenid Empire: Past Approaches, Future Prospects and Richard Taylor's The Greek Hoplite Phalanx: The Iconic Heavy Infantry of the Classical Greek have informed my views, along with several others that have had less impact on my thinking.

For Agincourt, in addition to the primary sources (Anne Curry (ed. and tr.) The Battle of Agincourt: Sources and Interpretations), I've been most heavily influenced by Anne Curry's 1415 Agincourt: A New History, Juliet Barker's Agincourt: The King; the Campaign; the Battle and Clifford J. Rogers' “The Battle of Agincourt,” in The Hundred Years War (Part II): Different Vistas, ed. L. J.Andrew Villalon and Donald J. Kagay. Robert Hardy's views on how deeply archers can be deployed is contained in The Great Warbow, by Matthew Strickland and Robert Hardy, which is also one of the most thorough works on the military history of the bow in England.

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^1 This is based on a file width of approximately one meter per man and a depth of four ranks. Tito Livio wrote that the English were "scarcely four deep", and this was not an unusual depth for the English men-at-arms. Robert Hardy has additionally argued that, if the English archers were to be able to shoot past each other (after the initial long range volleys), they needed to be only 3-4 deep when on flat ground, so this also fits the depth Livio mentions.

^2 5000 men-at-arms in the first battle, 3000 men-at-arms and 2000 gros valets in the second and the remaining gros valets (unknown, but anywhere between 2000 and 5000) in the third battle.

^3 This is somewhat different from more traditional versions of Agincourt, which don't suggest that some of the vanguard were already panicking and possibly routing by the time the English engaged in hand-to-hand combat, but is based on a passage jointly shared by the Burgundian chroniclers Monstrelet, Jean de Waurin and Jean le Fevre:

Those on horseback were so afraid of death that they put themselves into flight away from the enemy. Because of the example they set many of the French left the field in flight.

No other chronicle suggests that members of the vanguard began to flee the battlefield, but several others do mention extreme disorder and the inability of the vanguard to reform before the English archers attacked. Given the circumstances the vanguard faced, I think it's reasonable to suggest that they were at least panicking before the melee began.

From earlier answers about the Persian army, it seems like a bit of 1 and 2, though there is also the issue that we only have detailed accounts of Achaemenid tactics from a few battles against Greeks. And even at Marathon the Persians broke through in the centre of the line.

Our Greek warfare expert u/Iphikrates has written about this topic here and here (and in older threads like this, this and this. It has also been discussed a bit by u/dub_sar_tur in this and this answer. Sean Manning, who last year published an important monograph on the Achaemenid military, has written quite a lot about the topic on his blog as well. Some posts of his were linked in Dub_sar_tur's answers, and here he criticises the traditional schoarly view of the Persian army