Teacher always told me there's a difference but I still don't understand the difference. The textbook says "Melisma and lick have the same meaning but are appropriate to different styles of music". So what's the difference?
"Melisma" and "Lick" are essentially unrelated. A "melisma" is a series of notes all sung to the same syllable. A "lick" is a short musical idea, typically used in improvisational music, that can serve as a showy moment or as a motif within the music.
Where your textbook is accurate is that the two terms tend to be used to describe different styles of music. "Melisma" is used more in relation to classical music; "Lick" is used more in jazz, blues, and rock (i.e., improvised music). But they aren't exclusive to those styles, particularly "melisma".
Here is a video showing melismatic singing about a few contemporary, popular singers:
And here is some melisma in a classical setting:
Here are some licks...