This is part of quite a complex group of verbs that are, and have been throughout the history of the English language, frequently (con)fused in various ways.
The OED’s entrance on awake has a very thorough etymological description, which I quote here with some edits (removing extraneous details that obscure more than they clarify and highlighting a few things):
In this, as in the simple wake, two early verbs are mixed up; the form-history being complicated with that of awaken, as the sense-history is with that of awecche (q.v.).
For the intransitive verb, Old English has awæcnan, awōc, awacen, compound of wæcnan, wōc, wacen, the present stem having a formative -n-: wak-n-. This present began already in Old English to be treated as a weak verb, with past tense awæcnede; whence modern English awaken, awakened. Late Old English had also a weak verb awacian, awacode, in form a compound of wacian, wacode ‘to watch, keep awake’, but in sense identical with awæcnan, and perhaps originating in a confusion of the two. This gave Middle and modern English awake, awaked.
After the weak form awakened came into common use (as past tense of awaken), the original relation of awoke and its past participle to that verb became obscured; and later instinct, in accordance with the general analogies of the language, has referred them to awake, treating them as strong equivalents of awaked.
Of all these forms the sense was in Old English only intransitive ‘to arise or come out of sleep,’ the transitive (causal) sense of ‘rouse from sleep’ being expressed by the derivative awęcc(e)an, Middle English awecche (cf. German erwecken); but soon after 1100, awake began to be used in this sense also, and at length superseded awecche, which is not found after 1300.
There has been some tendency, especially in later times, to restrict the strong past tense (awoke) and past participle (awaken) to the original intransitive sense; and the weak inflection (awakened) to the transitive sense, but this has never been fully carried out.
The strong past participle awaken was already in 13th cent. reduced to awake, and at length became merely an adjective (mostly predicative), after which a new form from the past tense, awoken was substituted; but the weak form awaked is also in common use. (Shakespeare used only the weak inflections.)
Add to this what they have to say about awaken:
Old English awæcnan ‘to waken’. In Old English awæcnan was a strong verb with past tense and participle awōc, awacen. But sometimes the present stem (being irregular) was mistaken for a weak verb, whence already in 9th cent. the past awæcnede, modern awakened, which is now treated as the proper past tense, while awoke and its accompanying past participle are referred to the originally weak awake. Like awake, this was also at first strictly intransitive; the transitive use is of comparatively recent appearance, but now the most frequent.
– and we get a very muddled picture indeed. Especially when there is also the uncompounded verbs wake and waken to consider (they’re as convoluted).
My personal feeling, which corresponds quite well with the OED’s examples and description, is that both awake and awaken have the possibility to be used both transitively and intransitively, but that by far the most common usage is that awake is intransitive while awaken is transitive.
Moreover, awake is strong (awake, awoke, have awoken) while awaken is weak (awaken, awakened, have awakened).
In other words, I would say, intransitively:
I awake at six o’clock every morning.
I awoke at six o’clock yesterday morning.
I had already awoken when the alarm went off at six o’clock this morning.
– but transitively:
I awaken him at six o’clock every morning.
I awakened him at six o’clock yesterday morning.
I had already awakened him when the alarm went off at six o’clock this morning.
(However I say it, of course, it’s a big fat lie—there’s no way I’m awake at six in the morning. But that’s incidental.)
The opposite usages are historically well-founded, but (as mentioned in the highlighted paragraph in the etymology above) there has been a tendency to move away from them over recent centuries, and they often sound downright jarring to me, though not always. For example, “I awake him at six every morning” sounds quite ungrammatical to my ear, whereas “I was awoken at six this morning” sounds only somewhat ‘off’, and the Enya song I May Not Awaken sounds perfectly fine. (This is where the comments on the question that spurred this question become relevant: the usage there is transitive, as in “I awake him at six every morning”, and sounds downright ungrammatical to my ear.)
Of course, in actual, practical usage, I’m much more likely to use wake up (which is indifferent to transitivity) in both cases; but that’s irrelevant to the discussion about these particular verbs.