It's interesting, because this is one of the few moments we get from multiple perspectives. We get quite a bit of overlap from Sam/Jon with their early chapters in AFFC/ADWD, respectively. Here is the same line from Jon's chapter in ADWD:
Jon actually thinks about this again [SPOILERS ADWD] (/s "when he reads the Pink Letter.
Jon flexed the fingers of his sword hand. The Night's Watch takes no part. He closed his fist and opened it again. What you propose is nothing less than treason. He thought of Robb, with snowflakes melting in his hair. Kill the boy and let the man be born. (Jon XIII, ADWD)")
So Jon notes this connection himself, which just underscores its importance. You can also find it in Arya chapters:
Arya didn't know how much Robb would pay for her, though. He was a king now, not the boy she'd left at Winterfell with snow melting in his hair. (Arya VII, ASOS)
Sansa's chapters:
She had last seen snow the day she'd left Winterfell. That was a lighter fall than this, she remembered. Robb had melting flakes in his hair when he hugged me... (Sansa VII, ASOS)
So it's an interesting emphasis. Clearly it's nostalgic, as Arya/Sansa/Jon have this one memory as their last sight of Robb. However, it also seems to be useful in conveying the transition from being a boy to being a man. There is this sense that the leaving is absolute not just because the characters are likely never going to see each other again but because they're about to become someone else, maybe even unrecognizable, as Arya reflects.