why is it important to pack the UV islands as tightly as possible?
It is because you want to utilize as much space as possible. Let's say that we have 2048x2048 grid texture and we won't pack UV island tightly (there is a lot of space between islands):
You can see that a lot of UV space is wasted and the texture is not looking great on the robot. If i painted details with this uv layout everything would be blurry and pixelated. A lot more of the UV space can be occupied by packing the islands thighter, like here:
The grid tile texture became "smaller" on the robot what means it occupies more space on the uv now - there are just a few spots with a free space - it's impossible to fill the uv space 100%, but it definitely looks better.
The whole process of packing islands helps creating more detailed textures and just not wasting the uv space. But keep in mind we're talking about the textures that we will paint later on, not for procedural ones (you still can unwrap your model to have a procedural texture but usually object of generated coordinates are used).
So in essence, think about it as you don't want to be the Paris Hilton's of the unwrapping world; maladjusted, wasteful, squandering resources, unappreciative of hard work, no, you want to maximize the real state of the texture density. And Jim, if it turns out you're actually Paris Hilton trying to learn Blender under a fake name, I'm sorry, put that's the image the media paints of you. I think learning Blender will do you good, you go girl.