(At least in the US):
Get to know means to take one's time, make an effort or take special steps to acquaint yourself or familiarize yourself with someone or something:
I think she's interesting. I'd like to get to know her - her likes,
dislikes, beliefs, fears, hopes, etc.
Don't you think you should get to know one another before you get
Get to know your body and check yourself frequently for any discolorations,
bumps, spots, etc. that weren't there before.
New house? Get to know your neighborhood before you allow you children
to go out and play.
Come to know on the other hand means to come across information, receive information through happenstance, or eventually learn of something, and implies that there was a process involved:
How did you come to know that he was a spy? (=what series of events
led you to this information?)
Halfway through my research, I came to know that no such address ever
How did you come to know your husband? (= what led up to your meeting
him for the first time?)
This book will help you come to know God. (=through a process of
reading, introspection, prayer, bible study, etc. - a process that
will eventually lead to you knowing God).
The nuance of eventuality and process inherent in "come" can also be seen in the verbs: come to find, come to appreciate, come to the realization that, come to accept, etc.