Alpha (paid) and Classic (free) are completely different.
Classic is just a big sandbox where you can run around placing and destroying blocks. You are invincible, have infinite blocks of all types, so you don't need to gather resources or build tools or worry about survival. It's always day and there are no monsters. The point in this mode is just to build what ever you want. People can get pretty creative since it's a fully deformable environment. There's even a multiplayer mode where people can collectively work on building massive and amazing structures. The map in this mode is a limited size.
Alpha is a single-player "survival" mode that has a lot more features. You start with nothing and have limited abilities and health. You need to gather resources, build/craft tools and objects, mine for rarer but more valuable resources, and create shelter to survive the night. There is a day/night cycle and at night, monsters come out and try to kill you. In this mode, the map is "infinite", dynamically generating land as you explore. There are a lot interesting features to explore, like mountain ranges/valleys, underground cave systems. The ability to craft all sorts of interesting objects is also a big feature.
Here is a pretty neat fan-made trailer
http://www.youtube.com/watch?v=FaMTedT6P0I
TLDR: In short, Free (classic) is a pretty fun sandbox to just run around and build stuff, but Paid (alpha) is the main survival game mode with a lot of awesome features.