The primary difference is the state in which the nutrient is in. Nutrient broth's tend to be liquid, while agar plates tend to be solid. Although both can be composed of any number of nutrients depending on the wanted bacterial growth, the fact that broths are liquid and agar is solid manifest as different pros and cons when culturing bacteria. For example, agar plated cultures grow as flat cultures, favoring say animal epithelial cells which normally grow on a flat surface to which they attach, while bacteria grown in liquid cultures often form colloidal suspensions.