What is the difference between a pulse and a wave?

Both terms describe disturbances in some medium. Wave usually refers to a continuous disturbance. Like if you grab hold of spring and shake it back and forth a lot. Pulse, on the other hand, often refers to some type of one-time disturbance. Like shaking the spring only once.

Of course there will be overlap or ambiguities in these terms. I doubt there's any agreed-upon precise definition of these.

Waves is one common way in which nature expresses the flow of energy through space and time.

A pulse is just a special type of wave - a solitary wave. And solitary waves that dissipate a minor amount of energy into the medium, have low dispersion, and are able to maintain their shape over distance are called 'solitons'. A vortex ring is a good and interesting example of a solition.

Considering Fourier analysis, all types of waves, including pulses can be modeled by mathematics.