You could certainly construct a computing system that utilised either one of them as its only memory model. However, they both have rather different properties each with its own good and bad points. Most systems utilise both so as to get the benefits from each of them.
Stacks
A stack can be thought of as a pile of plates, you write a value on a plate and put it on the top of the stack this is called a push operation and stores a value on the stack. You can obviously also remove the top plate from the stack this is called a pop operation. But new allocations must always be at the top of the stack.
The stack tend to be used for local variables and passing values between functions. Generally stacks have the following awesome properties:
The problem with the stack comes from the fact items can only be added/removed from the top of the stack. Now this makes great sense when traversing up and down through function calls: pop functions inputs from the stack, allocate space for local variables on the stack, run function, clear local variables from the top of the stack and push the return value onto the stack. If on the other hand I want to allocate some memory and say pass it to another thread or in general free it far away from where it was allocated all of a sudden I have a problem, the stack is not in the correct position when I want to free the memory.
You could say the stack facilitates fast sequential memory allocation.
Heap
Now the heap is different each allocation is generally tracked separately. This causes a lot of overhead for allocations and deallocations, but each one can be handled independently of other memory allocations, well until you r