In “The Legend of Sleepy Hollow,” Brom Bones and Ichabod Crane are distinct in their looks, personality, and reputation: What does each man represent both in the story and as symbols of an American persona? Which character is arguably considered the “hero” of the story?

Washington Irving's "Legend of Sleepy Hollow" is a classic American folktale. Though originally published in 1820, its characters are familiar even today, for Ichabod Crane and Brom Van Brunt embody the dichotomies between town and country, intellect and labor, or more simply, brains and brawn, that are so well-established in American culture.
Within the context of the story, the men are rivals for the affections of Katrina Van Tassel, the most desirable young woman in the vicinity of Tarry Town. Brom was considered her undisputed favorite until Ichabod came to town to teach at the local school. Katrina's attraction to both men is the bone of contention between them, as each man feels his claims to Katrina's love are more worthy than his rival's. One could say that the value American society places on one character's virtues over the other's is analogous to Katrina's romantic whims and are just as unpredictable.
So what are each character's virtues? What can we say about them as archetypes within the American consciousness based on their personalities?
Ichabod is a stereotypical intellectual—a man whose learning and erudition must make up for his utter lack of physical prowess. Irving describes him thusly:

He was tall, but exceedingly lank, with narrow shoulders, long arms and legs, hands that dangled a mile out of his sleeves, feet that might have served for shovels, and his whole frame most loosely hung together. His head was small, and flat at top, with huge ears, large green glassy eyes, and a long snipe nose, so that it looked like a weather-cock perched upon his spindle neck to tell which way the wind blew.

Ichabod is an authoritative schoolmaster who keeps his students under strict control, but he is fair-minded and especially kind towards the weaker, more timid students, potentially because he himself used to be one:

Your mere puny stripling, that winced at the least flourish of the rod, was passed by with indulgence; but the claims of justice were satisfied by inflicting a double portion on some little tough wrong-headed, broad-skirted Dutch urchin, who sulked and swelled and grew dogged and sullen beneath the birch.

Like many people who cannot rely on physical charms to win the approval of others, Ichabod has developed a congenial personality and has "various ways of rendering himself both useful and agreeable" to the people of Tarry Town. He is kind and helpful, gentle with children, happy to volunteer, a good source of local gossip and, as someone well-travelled, a good source of news from further afield. He is not handsome, but he is clever and amusing, and Irving notes that:

Our man of letters, therefore, was peculiarly happy in the smiles of all the country damsels.

Intellectuals have a vivid mental life, which is stereotypically in contrast to their lack of a physical life. This vivid mental life goes hand in hand with a powerful imagination, and Ichabod is no exception to this trope:

He was, in fact, an odd mixture of small shrewdness and simple credulity. His appetite for the marvellous, and his powers of digesting it, were equally extraordinary; and both had been increased by his residence in this spell-bound region. No tale was too gross or monstrous for his capacious swallow.

The idea that intellectuals are divorced from the real world has a lot of traction in American society, and indeed throughout the world, for historically the people who could afford an education were not those who made a living through manual labor. There is a tendency to see highly educated people as residing in "ivory towers" and being "away with the fairies," i.e. not actively involved in the factual world. Ichabod's love of horror stories and his propensity for believing all of them is presented as a symptom of his personality—as a schoolmaster, he lives in and teaches his pupils about a world of theoreticals, and consequently he is more susceptible to believing things which "ordinary" people would dismiss as unrealistic.
For all that Ichabod is awkward-looking and gullible, however, he makes real headway with wooing Katrina Van Tassel, to the astonishment of Brom Van Brunt, who can see nothing of value in the man. His intelligence and kindly personality count for a great deal to Katrina, and while he is a highly credulous weakling compared to the pragmatic, athletic Brom, his weakness is its own sort of strength:

He had, however, a happy mixture of pliability and perseverance in his nature; he was in form and spirit like a supple-jack—yielding, but tough; though he bent, he never broke; and though he bowed beneath the slightest pressure, yet, the moment it was away—jerk!—he was as erect, and carried his head as high as ever.

The intellectual will come and go in the sentiments of the American people, but the key here is that he will come and go—and come again. His charms are less obvious than those of his counterpart, the strong, handsome, "ordinary" type, but they are perennial; they cannot ever be truly considered irrelevant to the American psyche. America has a difficult relationship with her intellectual class, for reasons far too complex to cover in this answer, but she never stops courting it.
Brom is exactly the type of "tough wrong-headed urchin" Ichabod disciplines in his classroom, the sort of person known in British society as a "lad" and in American society as "guy's guy." He is first introduced by Irving as Katrina's favorite suitor:

[...] a burly, roaring, roystering blade [...] the hero of the country round, which rang with his feats of strength and hardihood. He was broad-shouldered and double-jointed, with short curly black hair, and a bluff but not unpleasant countenance, having a mingled air of fun and arrogance.

Brom is handsome, popular, "Herculean" (i.e. extremely muscular) in build, loud, brash, and arrogant, but generally well-meaning, a kind of "rough diamond."

He was always ready for either a fight or a frolic; but had more mischief than ill-will in his composition; and with all his overbearing roughness, there was a strong dash of waggish good humor at bottom.

He is viewed in the region "with a mixture of awe, admiration, and good-will," for he typifies the personality of the good-natured "jock"—someone who is maybe a bit wild as a young man, but never malicious, and who wants only some maturity and "the love of a good woman" to settle him down into a fine, upstanding member of the community. He is the high-school football hero so often romanticized in American film and television, the ordinary man with ordinary dreams and a kind of rough-and-ready nobility. He is not concerned with the intellectual's hypotheticals; he lives firmly in the "real world," farming its crops, tending its flocks, raising its children, and safe-guarding its traditions. He never strays outside the acceptable boundaries of the community; in fact, he enforces them.
Brom is a native son of Tarry Town, while Ichabod is an outsider. This is important to an understanding of what these men represent in the wider context of American culture: the native sons of America are the sturdy, handsome, somewhat uncouth Broms, while the lanky intellectual Ichabods are outsiders who may temporarily capture the affections of American society but are bound to be driven out. Ichabods do not naturally spring from the soil of American culture; they must be artificially created through education. They have many charms and can be very useful and indeed driving them out of the small-town mindset of the American "community" does not eliminate them, but instead serves to further elevate them above the people of the Tarry Towns that make up the country. Ichabod, for instance,

having been suddenly dismissed by the heiress [...] had changed his quarters to a distant part of the country; had kept school and studied law at the same time; had been admitted to the bar; turned politician; electioneered; written for the newspapers; and finally had been made a justice of the Ten Pound Court.

He has not succeeded in winning Katrina's heart, but he is now a man of authority and high office. He could not organically become part of American society (embodied by the lush beauty of Tarry Town), so instead he rules over it. Brom, meanwhile, married the fair Katrina shortly after dispatching his rival and is not at all bothered by Ichabod's subsequent supposed elevation in life, for to his mind, he got the real prize. Ichabod—the intellectual—can keep his office and authority and book-learning; Brom—the ordinary guy—will take the material fortune of a beautiful (and wealthy!) wife and a profitable farm, and be content.
Who is the real hero of the story? Well, Irving does refer to Ichabod as "my hero" twice, both times in a rather tongue-in-cheek way, while describing Ichabod on his broken-down old nag, and Ichabod's joy at the sumptuous feast laid on at the Van Tassels's party. That aside, I don't know that it's possible to say which character is meant to be a "hero" in the story, because that assumes that Irving comes down on the side of bony, brainy, credulous Ichabod over handsome but overbearing Brom. Both men have their virtues and both have their flaws, and the tug-of-war between them for Katrina's heart is still played out in each American generation.