They've gone different ways since Sunnydale, parting in friendship, but without the bond they had shared. Buffy keeps telling Dawn to co-sign her name on the Christmas and birthday cards her sister sends Giles every year. Her life has stayed just as unsteady and unpredictable as when she was still The Slayer: no permanent address, no permanent job, no permanent relationship. She's not unhappy, though - just restless. It's not the same thing. Most of her time, she spends travelling (well, almost two decades later, the Council - or what's left of it - has decided to pay her at least the equivalent of Giles's salary during their time in Sunnydale), off to a different destination every few weeks - mostly within Europe.
One day, just having gotten on the train from Zurich to Paris, she spots Giles on the platform. It takes her a moment to recognize him. He hasn't changed much: his hair is of a silverish grey now, his posture slightly more fragile, his face a little slimmer, but his eyes are the same. And his smile, too: apparently, he's waving to a pretty girl around half Buffy's age - his daughter, maybe? She briefly considers jumping off the train and hugging him, but just then, it starts moving. Instead, she opens the window and starts waving and yelling his name, cheerfully at first, then almost sadly as the train is moving farther away.
She almost gives up hope, but he spots and recognizes her just in time. He waves back frantically, his smile so genuine that it breaks her heart. She regrets not having gotten off right away. Who said she was to be in Paris? She could just as well have spent another few days in Zurich. She starts crying as Giles is becoming more and more of a faint dot in the distance. As she leans back inside the train, she notices the young woman's eyes on her for a moment. The poor girl! She can't possibly have understood. Suddenly, a weird melancholia hits Buffy: a feeling of missed and irretrievable opportunities. This goddamn Paris.
She should really call Dawn and ask for his address.