We can use either "stay" or "keep" in the two following sentences. They mean the same thing. In the US I think "stay" is more common. "I wanted to see the Lyrids meteor shower, but I couldn't stay awake." "I wanted to see the Lyrids meteor shower, but I couldn't keep awake." In this next sentence, we can only say "stay." "I usually like it when the birds begin to sing in the springtime, but I hate when they start up at 4 am right outside my window. Sometimes they keep me awake."

"Keep awake" implies "keep yourself awake". This means you also have a passive form, "be kept awake". "Stay awake" doesn't have an object, so a passive form doesn't exist. Both expressions mean to remain awake, or continue to be awake. The main difference is in the grammar, not the definition.