To my perception, the difference is slight but there is one.
You can replace " ____ is illusionary " with " ______ is an illusion " and have nearly identical meaning. But with a difference of remove (and in some contexts), I’d instead replace " ______ is illusory " with " ______ is like an illusion "
So to me, illusionary means having the definitional quality of an illusion, while illusory can have that meaning, but it has another closely related sense, in which it means having some (perhaps connotational) qualities of an illusion, or being like an illusion (usually in addition to being an illusion, rather than instead of, but either is possible).
Calling a dream “illusionary” would be redundant, while calling a dream “illusory” would be descriptive. The latter would give an impression that the dream felt dreamlike - many dreams feel real while you’re dreaming them, but none actually are really happening.