Why do we need a Unit Vector (in other words, why do we need to normalize vectors)?

You don't have to normalize vectors, but it makes a lot of equations a little simpler when you do. It could also make API's smaller: any form of standardization has the potential to reduce the number of functions necessary.

Here's a simple example. Suppose you want to find the angle between two vectors u and v. If they are unit vectors, the angle is just arccos(uv). If they're not unit vectors, the angle is arccos(uv/(|u| |v|)). In that case, you end up computing the norms of u and v anyway.

As John D. Cook says - mainly you're doing this because you care about the direction, not the vector itself. Depending on context, you more than likely don't want / need the magnitude information - just the direction itself. You normalize to strip away the magnitude so that it doesn't skew other calculations, which in turn simplifies many other things.

In terms of AI - imagine you take the vector V between P1(the AI bad guy) and P2 (your hero) as the direction for the bad guy to move. You want the bad guy to move at a speed N per beat - how do you calculate this? Well, we either normalize the vector each beat, multiply by N to figure out how far they moved, or we pre-normalize the direction in the first place, and just multiply the unit vector by N each time - otherwise the bad guy would move further if it were further away from the hero! If the hero doesn't change position, that's one less calculation to worry about.

In that context, it's not a big deal - but what if you have a hundred bad guys? Or a thousand? What if your AI needs to deal with combinations of bad guys? Suddenly it's a hundred or thousand normalizations you're saving per beat. Since this is a handful of multiplies and a square root for each, eventually you reach the point where not normalizing the data ahead of time means you're going to kill your AI processing rate.

More broadly - math for this is really common - people are doing here what they do for things like 3D rendering - if you didn't unitize, for instance, the normals for your surfaces, you'd have potentially thousands of normalizations per rendering which are completely unnecessary. You have two options: one - make each function perform the calculation, or two - pre-normalize the data.

From the framework designer's perspective: the latter is inherently faster - if we assume the former, even if your user thinks to normalize the data, they're going to have to go through the same normalization routine OR you're going to have provide two versions of each function, which is a headache. But at the point you're making people think about which version of the function to call, you may as well make them think enough to call the correct one, and only provide it in the first place, making them do the right thing for performance.

You are often normalizing a vector because you only care about the direction the vector points and not the magnitude.

A concrete scenario is Normal Mapping. By combining light striking the surface and vectors that are perpendicular to the surface you can give an illusion of depth. The vectors from the surface define the parallel direction and the magnitude to the vector would actual make calculations wrong.