There's more to this, as well. It's a side we don't like to look at, but nonetheless, it's there.
We hurt when our loved ones die because humans are naturally selfish with regard to their emotions. When a loved one passes away, we realize that their lives will no longer be a part of ours. Their love won't touch our lives any longer, and we realize that we will quite possibly never see them again. (Totally dependent on your beliefs.)
This also causes people to get angry when a loved one dies. They invoke the rationale of "Why did he/she have to die? Why couldn't it have been someone else?" We say this, because that "someone else" doesn't mean as much to us as our loved ones. We're trying to protect our own emotions.
The state of "missing" someone is self-interest. They are not here with us, so we feel the loss of their being gone. It hurts our emotions, and we aren't as happy until they have returned. With death, it's more of a permanent situation. Therefore, the emotional pain is a lot more intense. We know they won't come back.