Why does Linux make use of tasklets (i. e., software interrupts) instead of executing all interrupt-related activity in the (hardware) interrupt handler?

Although some devices can be controlled using nothing but their I/O regions, most real devices are a bit more complicated than that. Devices have to deal with the external world, which often includes things such as spinning disks, moving tape, wires to distant places, and so on. Much has to be done in a time frame that is different from, and far slower than, that of the processor. Since it is almost always undesirable to have the processor wait on external events, there must be a way for a device to let the processor know when something has happened.

That way, of course, is interrupts. An interrupt is simply a signal that the hardware can send when it wants the processor's attention. Linux handles interrupts in much the same way that it handles signals in user space. For the most part, a driver need only register a handler for its device's interrupts, and handle them properly when they arrive. Of course, underneath that simple picture there is some complexity; in particular, interrupt handlers are somewhat limited in the actions they can perform as a result of how they are run.