take me back to the start
Many people were seemed really distraught that they didn’t hug, but, to me, their goodbye handshake, was more powerful than any hug. At the beginning, we can already see that the handshake is a significant to them both. It represented their start: professional, they were just two men—soon-to-be flat mates, and ‘end’—intimate, their relation exceedingly more vehement than before….and that’s something Sherlock doesn’t want to let go of. Notice the small tug in the second gif, you’ll see that the handshake also reveals additional details to Sherlock and John themselves. On one end, Sherlock, like the fans, expected something more, aware that after 6 months, the curtains will fall for him. Did he desire a hug? (A kiss?) One last ‘game’—the thrill of the chase, him and John against the world? (“Come with me.”)
Regardless of what Sherlock wanted, on the other hand, John refused it. He sees what’s behind Sherlock: his wife, his baby, the domesticity of family life…whether he subconsciously wants to move on or not, he knows he that this old life of his and Sherlock’s is drawing to a close. That this isn’t a life he can life live anymore. Thus he pulled back, as to say, “no, Sherlock, I can’t. (Not anymore.)”