What's the technical difference between fainting (and being unconscious) and falling asleep (and sleeping)?
For this answer I assume that we are talking about sleep in healthy individuals as well as fainting as an isolated event without underlying disease or injury.

The most common form of what we call "fainting" is vasovagal syncope. It is caused by a drop in blood flow to the brain (cerebral hypoperfusion) e.g. by (false) activation of the parasympathetic nervous system and inhibition of the sympathetic system, resulting in low blood pressure or drop in heart rate (reviews 1,2,3). This can be triggered by the Bezold-Janisch reflex. Syncope can also be a symptom of underlying diseases or a side effect of drugs.

The reduced blood flow results in a transient loss of consciousness, a 'shut down' of the brain, which we don't really understand. It is sometimes accompanied by a flattening of an EEG signal, even though the data doesn't seem to be very consistent. Consciousness is defined by two factors, arousal and awareness. The fainted person cannot be 'woken up' and is not aware of the situation (as you said, you don't remember the moment of fainting). But this state normally only lasts for a few seconds or 1-2 minutes.

Sleep, on the other hand, is a mechanism highly regulated by the brain, circadian rhythms and hormones. The brain is thought to switch between the states of sleep and wakefulness by an interplay of different sleep-promoting and wake-promoting areas, including the hypothalamus.

You can argue that sleep is not a state of unconsciousness, since dreaming or even lucid dreaming are connected to a conscious experience. While sleeping lasts longer than the unconsciousness during syncope, it can be reversed by an external stimulus. The sleeping person can be woken up, even though the arousal threshold is increased (loud noises from the alarm are needed). The brain, in fact, is highly active during sleeping (as can be seen in characteristic EEG signals), with energy use close to the state of wakefulness in REM sleep and about 85% during non-REM sleep.

Since these are two different mechanisms, you should not be able to switch between them. After fainting you should regain consciousness very fast. Activation of sleep mechanisms is unlikely in a healthy individual, where sleep regulation is normal and syncope is a rare event.