Wake up x Awake: Qual a diferença?

Se eu não estiver redonda ou quadradamente enganado, posso lhe adiantar que awake = wake up e que o primeiro vai ter uso mais formal, ao passo que o segundo, informal. Em outras palavras, na conversação você vai ver ouvir mais wake up, mas mui possivelmente, num relatório ou discurso formal, poderíamos esperar awake.

I awoke at six in the morning = I woke up at six in the morning

"Awake(verbo)" é formal. "Wake up" é a forma mais comum de se dizer "acordar" em inglês falado.

.: It was midday when she awoke.

.: James usually wakes up early

Não acho que grau de formalidade, mas sim a complexidade -- mesmo para os nativos, seja o ponto crítico. "Awake" é extremamente comum, e exerce outras funções, tal como no sentido de já estar acordado, ou em contato com alguma percepção:

I am awake. I woke up an hour ago.
I am awake to new feelings of happiness and gratitude.

Sim, é provável que se ouça mais "I woke up at seven" do que "I awoke at seven", mas a segunda forma não é incomum (sem preposição!).

E existe também a forma "to waken": "I wakened at seven." (Sem preposição, por favor!)

Essas formas requerem um pouco de estudo, examinando muitos exemplos, para descomplicar um assunto complicado mesmo. Do freeonline dictionary:

"waker n.
Usage Note: The pairs wake, waken and awake, awaken have formed a bewildering array since the Middle English period. All four words have similar meanings, though there are some differences in use. Only wake is used in the sense "to be awake," as in expressions like waking (not wakening) and sleeping, every waking hour. Wake is also more common than waken when used together with up, and awake and awaken never occur in this context: She woke up (rarely wakened up; never awakened up or awoke up). Some writers have suggested that waken should be used only transitively (as in The alarm wakened him) and awaken only intransitively (as in He awakened at dawn), but there is ample literary precedent for usages such as He wakened early and They did not awaken her. In figurative senses awake and awaken are more prevalent: With the governor's defeat the party awoke to the strength of the opposition to its position on abortion. The scent of the gardenias awakened my memory of his unexpected appearance that afternoon years ago.
Regional Note: Regional American dialects vary in the way that certain verbs form their principal parts. Northern dialects seem to favor forms that change the internal vowel in the verbhence dove for the past tense of dive, and woke for wake: They woke up with a start. Southern dialects, on the other hand, tend to prefer forms that add an -ed to form the past tense and the past participle of these same verbs: The children dived into the swimming hole. The baby waked up early."