What is the difference between kilesas and asavas? Or are they synonyms?
My view is 'kilesa' is the general or broad term referring the mental impurities or toxins.
Therefore, anusaya (underlying tendencies) are kilesa; asava (outflows) are kilesa; hindrances (nirvarana) are kilesa; and the various forms of tanha (craving), attachment & becoming (bhava) are kilesa.
The most subtle level of kilesa is anusaya (underlying tendencies). When the anusaya flow-out of the substratum of the mind, this flowing-out, leakage or fermentation ('bubbling-up') is called 'asava'. The anusaya flow-out as the five hindrances (nirvarana). If the five hindrances take control of the mind and push the mind to engage in external sense objects - craving, attachment & becoming will arise in relation to those sense objects.
In MN 9, the asava are included within the condition of ignorance in the dependent origination.
There are these three fermentations (asava): the fermentation of sensuality, the fermentation of becoming, the fermentation of ignorance. This is called fermentation. From the origination of ignorance comes the origination of fermentation. From the origination of fermentation comes the origination of ignorance.
Now that you have got your answer in the earlier post by Dhammadhatu, let me give you this little story from Supreme Buddha's life. How He recognized and overcame these to attain nibbana.
Supreme Buddha admits His past wanderings in existence which entailed suffering, a fact that evidently proves the belief in rebirth. He was compelled to wander and consequently to suffer, as He could not discover the Master Builder that built this house, the body. In His final birth, while engaged in solitary meditation which He had highly developed in the course of His wanderings, after a relentless search He discovered by His own intuitive wisdom the elusive Master Builder, residing not outside but within the recesses of His own heart. It was craving or attachment, a self-creation, a mental element latent in all. How and when this craving originated is incomprehensible. What is created by oneself can be destroyed by oneself. The discovery of the builder/architect is the eradication of craving by attaining Arhantship (perfect one), which in these verses is alluded to as “end of craving.”
The rafters of this self-created house are the passions (kilesa) such as attachment (lobha) aversion (dosa), illusion (moha), conceit (māna), false views (ditthi), doubt (vicikicchā), sloth (thīna), restlessness (uddhacca), moral shamelessness (ahirika), moral fearlessness (anottappa). The ridgepole that supports the rafters represents ignorance, the root cause of all passions. The shattering of the ridge-pole of ignorance by wisdom results in the complete demolition of the house. The ridge-pole and rafters are the material with which the builder/architect builds this undesired house. With their destruction the builder/architect is deprived of the material to rebuild the house which is not wanted. With the demolition of the house the mind, for which there is no place in the analogy, attains the unconditioned state, which is Nibbāna. Whatever that is mundane is left behind, and only the Supramundane State, Nibbāna, remains.