There are four species of Plasmodium that can cause malaria - falciparum, ovale, vivax and malariae. As you rightly point out, Plasmodium falciparum malaria is, all things being equal, regarded as the worst type to get. Why? Well, only this strain causes cerebral malaria (malaria affecting the brain) which is a major cause of death.
Falciparum manages to do this by adhering to blood vessels (vascular endothelium) throughout the body, but particularly in places like the brain, liver, intestine, etc. As the parasite matures inside red blood cells, it sends proteins to the surface of the erythrocyte, which are incorporated in its cell membrane. The erythrocyte consequently develops 'knobs' that protrude and are 'sticky' - they have a particular propensity to adhere to molecules on the endothelium. And this is when the fun starts.
These parasitised 'sticky' red blood cells start to clog up the circulation - particularly in the capillaries. In addition, the cells are less deformable the usual, which means that even without the stickiness they are more likely to get stuck in the capillaries. As if this wasn't enough, the sticky red cells can also stick to each other ('clumping') as well as to normal red cells ('rosetting'). Taken together, it is clear that the small vessels feeding our tissues get clogged up pretty quickly. And the inevitable result is hypoxia, lactic acidosis, and organ dysfunction.
If this happens in the brain, one can easily go into a coma, although there are other factors involved in this too. So it's the adherence of red cells to the capillary walls that is a unique consequence of falciparum in malaria, and that's what makes it so dangerous.