People often conflate the box under the TV with the service that connects it. They're fundamentally different things.
| Feature | Xbox 360 | Xbox Live |
|---|---|---|
| Type | Physical hardware | Online service |
| Primary Function | Runs games & media | Enables online features |
| Cost Model | One-time hardware price | Monthly/yearly subscription |
| Offline Use | Full single-player access | N/A (service requires internet) |
| Required For | Playing any game | Online multiplayer, cloud saves, free games |
Xbox 360 = The Car. You buy it once. It drives (plays games) on its own.
Xbox Live = Gas + Roads + Traffic Control. You pay regularly to access the highway system (multiplayer), fuel stations (digital Store), and traffic updates (friends, messages).
You need the console to play. You need Xbox Live to play online.