Many small consequences, some with class features, others with spells.
Mechanically speaking, yes, the only consequence of them having the Fey trait is that they cannot be affected by spells and effects that target humanoids, and can be affected by spells and traits that target fey. Humanoid-targeting features are a lot to list (some that come to mind are hold/charm/dominate person, lycanthropy, the warlock's Gaze of Two Minds eldritch invocation, calm emotions, crown of madness, or simulacrum), but below is a list of all the fey-related features.
Circle of the Land druids can't be charmed by Centaurs.
When you reach 10th level, you can’t be charmed or frightened by elementals or fey, and you are immune to poison and disease.
Oath of the Ancients paladins can turn the faithless:
Turn the Faithless. You can use your Channel Divinity to utter ancient words that are painful for fey and fiends to hear. As an action, you present your holy symbol, and each fey or fiend within 30 feet of you that can hear you must make a Wisdom saving throw. On a failed save, the creature is turned for 1 minute or until it takes damage.
They can be a Ranger's Favored Enemy (without having to pick two specific races of humanoid) and are revealed by their Primeval Awareness:
Choose a type of favored enemy: aberrations, beasts, celestials, constructs, dragons, elementals, fey, fiends, giants, monstrosities, oozes, plants, or undead.
For 1 minute per level of the spell slot you expend, you can sense whether the following types of creatures are present within 1 mile of you (or within up to 6 miles if you are in your favored terrain): aberrations, celestials, dragons, elementals, fey, fiends, and undead.
Arcana Domain clerics' Arcane Abjuration Channel Divinity option can affect fey:
As an action, you present your holy symbol, and one celestial, elemental, fey, or fiend of your choice that is within 30 feet of you must make a Wisdom saving throw, provided that the creature can see or hear you. If the creature fails its saving throw, it is turned for 1 minute or until it takes any damage.
You might be detected by commune with nature:
You instantly gain knowledge of up to three facts of your choice about [...] powerful celestials, fey, fiends, elementals, or undead
You are detected by detect evil and good:
For the duration, you know if there is an aberration, celestial, elemental, fey, fiend, or undead within 30 feet of you, as well as where the creature is located.
You are resisted by someone affected with dispel evil and good or protection from evil and good:
For the duration, celestials, elementals, fey, fiends, and undead have disadvantage on attack rolls against you
Forbiddance might harm you:
In addition, the spell damages types of creatures that you choose when you cast it. Choose one or more of the following: celestials, elementals, fey, fiends, and undead. When a chosen creature enters the spell's area for the first time on a turn or starts its turn there, the creature takes 5d10 radiant or necrotic damage (your choice when you cast this spell).
Hallow stops you from entering, unless the caster makes an exception:
First, celestials, elementals, fey, fiends, and undead can't enter the area, nor can such creatures charm, frighten, or possess creatures within it. Any creature charmed, frightened, or possessed by such a creature is no longer charmed, frightened, or possessed upon entering the area. You can exclude one or more of those types of creatures from this effect.
Magic circle can affect you:
Choose one or more of the following types of creatures: celestials, elementals, fey, fiends, or undead. The circle affects a creature of the chosen type [...]
You can be bound to serve someone with planar binding:
With this spell, you attempt to bind a celestial, an elemental, a fey, or a fiend to your service.
You can be left out of a temple of the gods:
The temple opposes types of creatures you choose when you cast this spell. Choose one or more of the following: celestials, elementals, fey, fiends, or undead. If a creature of the chosen type attempts to enter the temple, that creature must make a Charisma saving throw. On a failed save, it can’t enter the temple for 24 hours. Even if the creature can enter the temple, the magic there hinders it; whenever it makes an attack roll, an ability check, or a saving throw inside the temple, it must roll a d4 and subtract the number rolled from the d20 roll.
Some spells force a fey back to its homeplane, but I don't think Centaurs have a different home plane, so ignore those. With a home-brewy DM, you might be able to conjure a centaur with conjure fey, or conjure woodland beings.