The most distinct difference is that most curry pastes use fresh components such as fresh herbs whereas dried curry powder is just that, dried. If you compare a Madras curry powder to Madras curry paste you'll find that the flavor of the paste is more intense. Curry powder can quickly lose flavor and you'll notice that most Indians create and grind their powders almost daily. Toasting the spices before making the powder can develop the flavors. Curry paste is ready to use because it already contains ghee or other suitable oil. With powder you'll have to add that.
When looking at the differences between Thai curry powder and Thai curry paste the differences are somewhat similar. The paste is generally ready to use and contains fresh lemongrass, oils, and sometimes coconut milk.
What I use depends on what I am making. Sometimes I will bring the paste on the trail because it works out the same weight-wise if I have to carry extra fats for the powder.
In most cases, the flavor of the pastes are better for some recipes. It's a preference thing and highly individual. I'd recommend trying both (at home) and seeing which you like more.