This is rather a knotty little group of verbs that have gone all over the place in Modern English, though they were very clearly and regularly distinguished in Old English.

Let’s start with the etymology, and then move on to current usage.

## Etymology ##

The words _watch_, _(a)wake_, _(a)waken_ all share a common root. The Proto-Indo-European root was _*u̯eǵ-_, and its basic meaning approximately ‘to be strong/quick/lively/aroused/powerful’. It underlies for example Sanskrit वाजः _vā́ jaḥ_ ‘power/force’, Latin _vigil_ ‘watchful/awake’ (as in English _vigil_ also), _vigere_ ‘be lively’ (as in _vigorous_ in English), _velox_ ‘fast/lively’ (from _*veg-lox_; as in _velocity_ in English).

In Proto-Germanic, the root regularly became _*wek-_, with the ablaut form _*wak-_. This latter form generally took over, because it was found in a very common construction with this root: the causative (PIE) _*u̯oǵ-ei̯e-_ > (PG) _*wak-ja-_ ‘make lively’ = ‘awaken [someone], wake [someone] up’. So for all practical purposes, the Germanic root was _*wak-_.

There were then at least three different verbs derived straight from this root present in Proto-Germanic (note: _*-aną-_ is the reconstructed Proto-Germanic infinitive marker):

 1. A simple verb _*wak-aną-_, retaining the original meaning of the basic root, ‘be lively/awake’
 2. A nasal inchoative<sup>1</sup> _*wak-n-aną-_, meaning ‘become awake’ = ‘wake up [by oneself]’
 3. The causative _*wak-j-aną-_ mentioned above, meaning ‘make lively/awake’ = ‘wake [someone] up, awaken [someone]’

These were all quite regular, since both the causatives and inchoatives were regularly formed in Germanic by adding the _*-j-_ and _*-n-_ suffixes.

In Old English, they were also relatively distinct, although some parts of the paradigms had already started to coalesce. Most notably, both the old base verb and the old inchoative verb had become strong verbs in verbal class VI. This is original to the base verb (which is a primary, non-derived verb and therefore shows ablaut—the Proto-Indo-European equivalent of Germanic strong verbs), but not to the inchoative verb. The inchoative verb is a derived one, and it ought to have been weak; but it had fallen together with the base verb in the past tense.

This coalescence is very clear if you look at the Wiktionary paradigms for [Old English _wacan_](http://en.wiktionary.org/wiki/wacan#Old_English) and [Old English _wæcnan_](http://en.wiktionary.org/wiki/w%E6cnan#Old_English): the past tense forms are the same, the others are distinct.

The causative _*wak-j-aną-_, on the other hand, had maintained its different conjugation throughout: it remained a weak verb of class II, [Old English _wacian_](http://en.wiktionary.org/wiki/wacian#Old_English).

So at a brief glance, if we give just the third person singular of the three verbs in Old English:

 1. _Wacan_: present _wæceþ_, past _wōc_
 2. _Wæcnan_: present _wæcneþ_, past _wōc_
 3. _Wacian_: present _wacaþ_, past _wacode_

In Middle English, when final syllables started being reduced and lost, the differences between these three paradigms started getting muddled. The Middle English infinitive forms are the ones you have found on Wiktionary yourself:

 1. _Waken_
 2. _Waknen_
 3. _Wakien_

Once the forms started getting muddled, obviously the meanings started to overlap—it’s difficult, after all, to maintain a difference between ‘becoming [by one’s own accord] awake’ and ‘causing [someone] to be awake’ and ‘being awake’ when half of them have the same shape in various forms in the paradigm.

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## Current usage ##

In current English, there are only two basic forms left: _wake_ and _waken_.

_Wake_ is, in form, basically from the old base verb _waken_ < _wacan_. The modern verb _waken_, on the other hand, is from the inchoative _waknen_ < _wæcnan_. Note that in both these words, it is the final _-en_ that has been lost over time, and modern _waken_ thus represents Middle English _wakn_, to which of course an extra little prop vowel was added, since /kn/ is not a valid way to end a syllable in English.

In Middle English (as in Old English), both _waken_ and _waknen_ generally had a strong past form, _wook_ (in various spellings), which is what gives modern _woke_. The weak causative _wakien_ had a weak past, _waked(e)_. When the form of the causative was mostly lost (merging with the base verb once _-en_ disappeared), the weak past was kept in some dialects, the strong past in others, and both in yet others.

On the other hand, the inchoative _waknen_ developed a new weak past of its own, _wakned(e)_ and stopped using _wook_ like the base/causative verb. So there was a shift in what verbs shared what past tense forms:

 - [base + inchoative] strong past _vs._ [causative] weak past >  
[base + causative] strong or weak (or both) past _vs._ [inchoative] weak past

This is the reason why _waken_ today has only a weak past form, _wakened_, while _wake_ can have both strong _woke_<sup>2</sup> and weak _waked_.

When the causative disappeared as a separate form, the rather unusual thing happened that its semantics were taken over not only by the base verb (with which it merged in form), but _also_ by the inchoative verb. This means that both the base verb and the inchoative verb can now be either transitive or intransitive, which is a messy situation, but reality.

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In current English, _woke_ is the standard past tense of _wake_, both transitive and intransitive, causative or not; _waked_ is marked as nonstandard (dialectical) or archaic, and it’s nowhere near as common as _woke_.

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<sup>1</sup> There’s quite a bit of discussion about what the exact usage of these nasal formations were in Common Germanic. They are often based on nouns describing people or jobs, and they then mean ‘do/behave as X does’, a basically causative or factitive meaning; thus, in Old English, we get _læcnian_ ‘heal’ from _lǣċe_ ‘doctor’. But they are also used more or less as inchoatives from adjectives, as _blacken_ ‘become black’ from _black_.

<sup>2</sup> As Jon mentions both in his answer and a comment here, _woke_, phonemically /woʊk/, is not quite regular from the Old English _wóc_. Some analogy with similar-sounding verbs like _break_, _speak_, and _steal_ (which did sound the same earlier on, though they’re different now) took place at some point. Middle English _wook_ or _wooke_ or _woke_ can be regular from _wóc_, assuming that they represent /wuk/, which is what the Old English form should give—but that’s not necessarily a given, considering the vagaries of Middle English spelling.