What is difference between Nibbana and asaṅkhataṃ (unconditioned)?

A lot of sankharas are around unconditioned. Sannas (perceptions) are anicca (impermanent), good householder. Not out of no reason the Sublime Buddha encouraged his disciples to even let go of perception of Nibbana, at least (a talk where the monks didn't rejoice at the end, and to assume that good householder does so like-for-like).

What's the purpose of this thought sankharas, those buildings which wouldn't last? Focus on abandoning akusala and instead nourish kusala, when undertaken on trust, would lead one beyond. Ditthis (views) are only useful if taking on sammaditthi (right view), so let go of all else.

Nibbana is that which the mind experiences when it is free of defilements, free of the three poisons of greed, aversion and delusion. It is not a state of mind or consciousness. It is certainly not a self or a witness. Nibbana is not conditioned as in it is not caused or created by anything, or composed of anything, or influenced by anything. On the other hand, states of mind or consciousness is conditioned and changing as time goes on.

AN 9.34 states that nothing is felt or sensed about Nibbana and that's precisely why it is blissful. The absence of suffering (dukkha) is blissful. It's not blissful because it is felt by the senses to be pleasurable - that would make it conditioned. This answer explains more on Nibbana and bliss.

According to Iti 44, Nibbana-element without residue left describes the five aggregates in parinibbana, based on Ven. Thanissaro's footnote. This means that all five aggregates, including consciousness, stop operating completely. In this case, all mental fabrications (sankhara) have been stilled.

Nibbana-element with residue left describe the the five aggregates of a living Arahant i.e. the five aggregates (form, feeling, perception, consciousness, mental formations) are functioning without craving, clinging and defilements. The flames of passion, aversion and delusion are no more burning for the living Arahant. But the five aggregates function without clinging, like glowing embers, for the living Arahant. In this case, only the clinging mental fabrications (sankhara) have been stilled. This question discusses clinging aggregates vs. non-clinging aggregates. In parinibbana, the glowing embers are put out (Nibbana-element without residue left).

So, you can see that that when the Arahant is living, the clinging sankhara (mental fabrications) are stilled. In parinibbana, all sankhara (mental fabrications) are stilled.

What's interesting is that all mental fabrications (one meaning of sankhara) are conditioned (another meaning of sankhara), but not all that is conditioned are mental fabrications. For e.g. water is conditioned, but it's not a mental fabrication (unless you refer to the mental concept of water).

So, in my opinion, the unconditioned (asaṅkhataṃ) in Ud 8.3 may refer to all that is unconditioned. But "the stilling of the conditioned is bliss" (saṅkhārūpasamo sukho”ti) from Iti 43, specifically refers to clinging mental fabrications and not the stilling of everything that is conditioned.

The born, come-to-be, produced,
The made, the conditioned, the transient,
Conjoined with decay and death,
A nest of disease, perishable,
Sprung from nutriment and craving's cord —
That is not fit to take delight in.

The escape from that, the peaceful,
Beyond reasoning, everlasting,
The not-born, the unproduced,
The sorrowless state that is void of stain,
The cessation of states linked to suffering,
The stilling of the conditioned — bliss.

Iti 43 (translated by Ireland)

Interestingly, Ven. Thanissaro used the word "fabrications" instead of "conditioned" in his translation. I would go further to call it "clinging fabrications".

The stilling of the conditioned (i.e. clinging mental fabrications) is bliss.

The born, become, produced,
made, fabricated, impermanent,
composed of aging & death,
a nest of illnesses, perishing,
come from nourishment
and the guide [that is craving] —
is unfit for delight.

The escape from that is
calm, permanent,
beyond inference,
unborn, unproduced,
the sorrowless, stainless state,
the cessation of stressful qualities,
the stilling of fabrications, bliss.

Iti 43 (translated by Thanissaro)

So unconditioned is not right meaning of asaṅkhataṃ ... Ideally, it should be unconditionable? What do you say?

Unconditionable can be understood too easily as applying to a self. It is therefore not an ideal translation.

Unconditioned actions are unfettered by notions of a self.

For the living, there is still conditioning (e.g., one eats to sustain the body).

MN121:12.3: There is only this modicum of stress, namely that associated with the six sense fields dependent on this body and conditioned by life.’

That living conditioning disappears on relinquishing the body .

DN16:3.7.2: “parinibbātu dāni, bhante, bhagavā, parinibbātu sugato, parinibbānakālo dāni, bhante, bhagavato.
DN16:3.7.2: “Sir, may the Blessed One now become fully extinguished! May the Holy One now become fully extinguished! Now is the time for the Buddha to become fully extinguished.

This how i understand it,

The designation for Nibbana is 'removal of greed, anger & delusion.' It has a variety of qualified usage.

We say that the attainment of Arahantship is a removal and that an Arahant is one who has removed from the mind. The mind of an Arahant is inherently conditioned, is a sankhara but it has no greed, anger & delusion. We say that the attainment of Arahantship is an attainment of Nibbana in this sense of removal.

It is said that removal depends on destruction and that these unwholesome qualities utterly cease in one who practices the signless release or the four types of mindfulness as satipatthana leads to the signless release.

The signless release is an entering into & emerging from a meditative attainment of the 'cessation of perception & feeling'. It is also apprehended as emptiness or undirected, depending on how it was approached and what meditative theme was dominant in course of development.

This attainment is a direct seeing with the dhamma eye, one is then absorbed based on this nibbananirodhadhatu [extinguishment-cessation-principle] and it is said that this sorrowless [asoka] state is empty of greed, anger & delusion.

It is said that greed, anger & delusion are a making of signs. Therefore this attainment of signless release qualifies to be called a removal of greed, anger & delusion and a destruction because it is empty of those qualities and because these eventually utterly cease in who practices it.

When talking about the cessation of perception & feeling one could say 'it's the foremost extreme pleasure' and in regards to that one could say Nibbana is that foremost pleasure where nothing is felt.

In this sense one would say Nibbana [removal of greed, anger & delusion] is pleasant and have it refer to the Asankhata element which is Asoka [sorrowless].

When studying the words Deathless and Nibbana we have to keep in mind three primary circumstances

1. One who has delusion [non-arahant] prior to entering and emerges with delusion [non-arahant]
2. One who has delusion [non-arahant] prior to entering and emerges without delusion [arahant]
3. One who has no delusion prior to entering and emerges likewise.

All three absorbtions occur based on nibbananirodhadhatu which is empty of GAD but are going to be spoken about very differently. Eg we don't say that an arahant is destroying taints because he has none prior to entering and we don't say that a person who emerged from cessation of perception & feeling without removing all taints is one who has attained removal of taints in the sense of attaining arahantship.

There is a bit more to it but it's close. For full analysis one also needs to delineate differences between 'nibbana with residue', '..without residue/fuel', 'parinibbana' and 'unprovoked release'.