The crucial difference is that type aliases are assignment-compatible with their underlying type (and with other type aliases with the same underlying type), while inline classes are not.
In other words, inline classes introduce a truly new type, contrary to type aliases which only introduce an alternative name (alias) for an existing type:
typealias NameTypeAlias = String
inline class NameInlineClass(val s: String)
fun acceptString(s: String) {}
fun acceptNameTypeAlias(n: NameTypeAlias) {}
fun acceptNameInlineClass(p: NameInlineClass) {}
fun main() {
val nameAlias: NameTypeAlias = ""
val nameInlineClass: NameInlineClass = NameInlineClass("")
val string: String = ""
acceptString(nameAlias) // OK: pass alias instead of underlying type
acceptString(nameInlineClass) // Not OK: can't pass inline class instead of underlying type
// And vice versa:
acceptNameTypeAlias(string) // OK: pass underlying type instead of alias
acceptNameInlineClass(string) // Not OK: can't pass underlying type instead of inline class
}