The difference is that 'enthrall' suggests fascination/captivation/being under a spell, whereas 'enamor/enamour' suggests falling in love. The similarity is that these are both words which are pretty unusual in everyday English, and that neither is worth your while worrying about too much.

They can be used in the same way. Usually, "enamor" carries more of a sense of one's heart is moved, often with romantic implications. Both are normally used in the passive form, eg "I am enamored by that sweet girl" and "John was enthralled by the symphony's recent performance."