The Fault in Our Stars is technically set in Indianapolis, but it's not the Indianapolis of tourists or even of normal residents who go to work and school each day. It's the Indianapolis of the sick.
For Hazel and her companions, their hometown revolves around the places that they can frequent as cancer kids: hospitals, support groups in churches, and occasionally each other's homes. The whole setting is quite stifling and very indoors. After all, Hazel spends most of her time in her room reading the same book over and over again. It's pretty claustrophobic.
Hazel's totally aware of it, too. She describes Indiana as someplace where she feels confined: "It was a cloudy day, typical Indiana: the kind of weather that boxes you in" (4.48). Even when she meets Augustus and they start doing fun and exciting things together, they're still limited in where they're allowed to go: they go to each other's homes and to support group; they read a lot, watch TV, and hang out with their hovering parents; they spend time with Isaac, who also has similar limitations.
The Indianapolis that they see is one that is sterile, confining, and utterly familiar.
Ah, Amsterdam. The name itself conjures up visions of cobblestone streets, canals, bicycles, quaint coffee shops, artists, and of course, romance. So you know, basically the polar opposite of the Indianapolis we just left. When Hazel and Augustus go to Amsterdam, it's not just another country or continent for them. It's a totally different life, one that doesn't involve regular trips to the hospital and rolling their eyes as they sit in the "literal heart of Jesus":
It looked nothing like America. It looked like an old painting, but real—everything achingly idyllic in the morning light—and I thought about how wonderfully strange it would be to live in a place where almost everything had been built by the dead. (11.9)
These two kids are catapulted into a world where anything is possible, where they can have a romantic dinner outdoors, climb flights of stairs, and even sneak into each other's hotel rooms for sexy times. Even Hazel's mother, who is usually hovering right by her, acts differently when they're in Amsterdam; she allows Hazel and Augustus to go off on their own and manages to enjoy her own time as she tours Amsterdam.