Q:

"Smiled at me" vs. "smiled to me"

I searched for the following two expressions and came across both being used in various places. 

She smiled at me at the grocery store. 
She smiled to me at the grocery store. 

What is the difference between the two? 

A:

"Smile at" is more common, so saying "smile to" is a way to indicate a more "decisive" smile, or intent to communicate via the smile.  
I do not understand why others say "to" is incorrect here.  It is less common, to be sure, but that does not make it incorrect.  You can wave to a crowd, or wave at a crowd...but just because we say "wave to" more often, that does not make "wave at" incorrect.