Conoscente is:
a) somebody whose existence you are aware of
b) somebody that you will recognize when you meet him or her and say hello to.
Amico is:
a) he must obviously be a 'conoscente'.
b) somebody whose company you enjoy.
c) somebody that you will help if need arises, and this must be reciprocal.
d) somebody you trust, and he trusts you
== Just for completenes, and because I ebnjopy writing in English:
Usually /conoscente/ and /amico/ is bidirectional (symmetric behaviour), while, for instance /being fond of somebody/ is usually not a symmetric behaviour.
Wish you find good and sincere friends!