Do illusions exist as images and sounds in the world, like holograms, or do they exist only in the beholder's mind, like mass hallucinations? There are paragraphs in the PHB that can support both assumptions.
Clarification: the question primarily concerns illusions produced by Minor Illusion, Programmed Illusion, Silent Image, Major Image, Disguise Self spells. Spells like Fear or Phantasmal Killer explicitly says they affect creatures, so they are out of the scope.
Illusions as actual images and sounds
Spell description explicitly says that the caster "creates a sound or an image":
You create a sound or an image of an object within range
When casting an illusion, the caster have to specify a location, not a creature. Anyone who looks at the location perceives the illusion.
Many obviously mind-affecting spells like "friends" or "charm person" are enchantments, not illusions.
While there are creatures immune to charm, there is no creatures immune to illusions. Even a construct can perceive an illusion.
Illusions as mind tricks
The wizard's arcane tradition describes the School of Illusion as
magic that dazzles the senses, befuddles the mind
Being revealed, an illusion looks different to the perceiving person only:
If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
When does it matter
There are several cases when the outcome might depend on where the spell effect is located:
Antimagic Field – to be negated, should the illusion "itself" be in the field, or does any creature in the field become immune to illusions?
Detect Magic – assuming that Detect Magic spell can detect illusions, what exactly should be in the 30 feet range to be detected?
Also there are ambiguity with light, vision and line of sight:
Block light – A caster tries to use an illusion to cover a window, or conceal a lit torch.
Block vision – Many cases. A caster tries to hide an object "under" an illusion, but the spell description says it can only "create an image" and not "hide an object". If we assume an illusion actually can cover an object or a creature, Disguise Self would protect from all LoS-dependent spells, etc.