Dispel magic causes it to fade; antimagic field causes it to immediately disappear while it's in the field's area.
Dispel magic ends spells:
Choose any creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends.
Ending the spell will cause the steed to fade out over 1 minute.
Antimagic field is different: it doesn't end spells, but suppresses their effects:
Within the sphere, spells can't be cast, summoned creatures disappear, and even magic items become mundane. Until the spell ends, the sphere moves with you, centered on you.
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it.
A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.
So if you ride your phantom steed into an antimagic field, the steed will instantly cease to exist and you'll be forcibly dismounted. In principle, if the source of the field moves away, you should get it back as the spell is no longer suppressed.
It's somewhat questionable whether the steed itself counts as a creature created by magic. Phantom steed is an Illusion spell, the horse is described as "quasi-real", and the steed fades out of existence if it takes damage. (Contrast the Paladin find steed spell, which is Conjuration and summons a real creature which can take damage normally.)
This shouldn't affect its response to antimagic field because it suppresses spell effects of any kind. But it does matter for other spells like hallow and magic circle which could be used to restrict a Paladin's steed or a familiar.