Please help to explain the difference between these 3 terms:
They were translated here by Ven. @sujato and also by Bhikkhu Bodhi correspondingly as: conceit, arrogance and vanity.
I would like to understand more thoroughly the contrast difference and also the specific context in each case that those terms are being used. Thank you. 
New Concise Pali English Dictionary
ati
prefix
- prefix to nouns, adjectives, adverbs; preverb to verbs; before vowels (except ī-) generally, but not always, taking the form acc-
- as prefix to nouns expresses: abundance, excess, superiority, transgression; the being beyond (the reach of)
- as prefix to adjectives, adverbs expresses: very much, too much
- as preverb to verbs and their derivatives expresses: beyond, over; through; too much, very much, too far
- ati is sometimes inserted in iterative compounds expressing: and, adding further, even more, etc.
I recommend to read the suttas here SuttaCentral (such as Thag 6.9, AN 6.106, MN 54, Snp 4.10, Snp 4.15), which contain the relevant words, such as ‘atimana’, in an extended narrative. 
For example, MN 135 includes “atimāna” plus describes what the relevant behaviour arising from atimāna is like :
Take some woman or man who is obstinate and vain.
Idha, māṇava, ekacco itthī vā puriso vā thaddho hoti atimānī—They don’t bow to those they should bow to. They don’t rise up for them, offer them a seat, make way for them, or honor, respect, esteem, or venerate those who are worthy of such.
abhivādetabbaṁ na abhivādeti, paccuṭṭhātabbaṁ na paccuṭṭheti, āsanārahassa na āsanaṁ deti, maggārahassa na maggaṁ deti, sakkātabbaṁ na sakkaroti, garukātabbaṁ na garukaroti, mānetabbaṁ na māneti, pūjetabbaṁ na pūjeti.
Therefore, the description in MN 135 above, for example, shows a person with ‘atimana’ is not simply ‘conceited’ but very rude, very disrespectful and ‘excessively conceited’. 
Ati = excessive
Mana = conceited