What is the difference between kilesas and asavas? What is the difference between kilesas and asavas? Or are they synonyms?
Good question. 

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 &amp; 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 &amp; 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. 
  
  http://www.accesstoinsight.org/tipitaka/mn/mn.009.than.html