Historically there was no distinction between these words. “Buddha” means “awakened”. The term “enlightened” started to appear much later in the literature, as a synonym, and slowly replaced it. Then the term “awakened” rose back into prominence when spirituality became commoditized and distilled, because to most spiritual people “enlightenment” still felt like something much more or even an impossible goal, to which they are on an infinite journey.

Gautama would not consider the vast majority of people who nowadays claim to be awakened/enlightened to be awake, and neither do I.

Gautama Siddhartha lost his ego when he became awake, it disappeared, forever. Nowadays the term “awake” refers to an “awakened ego”, that is, a person who has awakened to the fact of non-existence of their ego, but hasn’t lost it yet. So, the ego awakened to its own non-existence, but is still largely present. This gives rise to the need for descriptions like “see through ego”, “shadow side”, “my true/higher self” and alike, such wordings are a relatively new phenomenon.

Gautama would never use this term like that. The distinction between the words “awake” and “enlightened” did not exist at his time. Even when he trained his Bodhisattvas who already realized emptiness, which is much more than can be said about most people who are currently “awake”, he referred to them as “unawakened Bodhisattvas”. So he could have call such people something else, but not “awake”. Maybe that’s where the term “stream-enterer” originated from in Buddhism.