They are distinctly related/similar. No one is suggesting otherwise. They both effect the mind to varying degrees. Seems people have been quite clear...and you'rejust refusing to accept a difference.
The one deals in the senses...perceptions, figments, images and sound, light, shadow, color. Can it deal in or change "minds" and/or "feelings," sure. But it does so by tricking the mind or feelings with sensory [illusions] and extra-sensory [phantasms] perceptions.
The other deals in minds and feelings, period...Altering them. Attacking them. Can it deal in images and light and sound? Sure. It can. But it doesn't have to.
In my homebrew, I make no distinction between "Illusionist" mages and "Enchanter" mages [or, rather, what is called an "Enchanter" in world is not just "a mage who focuses in enchantment magic."], but there is a distinction between Illusion Magic -"the Art of Imagining"- and Enchantment Magic -"the Art of Feeling."
There are very few spells of any school that are only and entirely composed of a single, shall we say, "frequency" of magic. The composition and proper casting of spells require a magic-worker to understand the "calculations" and "formulae" of combining all of the ingredients properly...or you don't get a spell effect. It's part and parcel of the reason Joe Farmer who hears and sees the PC mage waves his hands and say "Ibilis inik tangelo" can't just get a group of goblins to fall asleep by doing the same thing. It takes years of training to figure out how to take ABC glyphs, syllables, shards of comprehension from THIS part of "Arcane Magic," and putting it in/with/around the XYZ from THAT part of Arcane Magic and add them together into the right motion and syllables and designs to end up with "gigantic explosion of fire."
MOST "Illusionist" spells are illusion (whether or not they are also a phantasm). Many also combine with enchantment effects, and several overlap into conjuration (generating materials or energies which).
Hypnotic Pattern...is it an illusion or an enchantment? There is a swirl of luminous color (illusion) that hypnotizes the onlookers (enchantment). The argument could also be made that the generating of that swirling pattern -whether you wish to flavor it as "colored light, rainbow mist, energetic plasma," whatever stuff- is conjuration (generating matter/independent things) and/or evocation (generating energy). But, ultimately, the spell effect is a visual image effect that alters the mental state of its target(s) = illusion & enchantment.
One is light, color, shadow, darkness, fooling and/or tricking visual/audial/mental perceptions. "I see a rainbow pattern swirling and undulating over and over and..." While, in reality, there is nothing "real," solid, tangible, there.
One is just straight up direct "usurping" (for lack of a better term) and/or dominating thought, emotion, psychology. SEEING the TRICK [magic rainbow pattern] of the illusion, "It's so lovely...I just want to watch...So pretty...I just want...I ju-<drool, blank stare>..." THAT is all going on in the target's head. Directly scrambling their mental state...the enchantment.