What's the difference between normal cabbage and brasied cabbage?

Braising is a form of cooking. The cabbage itself is the same. You can eat cabbage raw, boiled, steamed, sauted, grilled, roasted, or braised. As Gypsfish noted, braising is cooking in a oven at a lower temperature with some type of liquid. It is the preferred cooking method for brisket, as an example.

One is normal and one is braised.

Braised cabbage has been cooked slowly in some kind of liquid, often with seasonings or some meat. It makes it very tender and juicy.